You have to go back to Newton De Moivre and Cotes And Wallis to understand the origin of measurement in Quantumn Mechanics.
The Era of Napierian Logarithms and the great Table calculations provided a core set of skills and direction for a subject just hot of the press called mathematics. Although Mathematicians like to go back and change history, there was no subject called mathematics until about the 15th century and even then it meant Gematrial thinking in regard to applying Pythagorean, Platonic and Euclidean principles to the study of God's creation, or Nature. This was called Natural Philosophy and metaphysics and was a curriculum devised by the Aristotelian schemas. There were no real hard boundaries between the declared subject areas and many areas were remodeled and renamed.
Thus Mathematics at Cambidge was only established in Newton's Time under a Curriculum imported and set out by Isaac Barrow. At the same time Wallis and all Scintists were ardent Euclidean Geometer's. And Descartes methods and Coordinate methods and praxis were the new thing being studied. Algebra, as laid out by Al Kwharzim and europeanised by bombelli and then Descartes and DE Fermat was not a separate subject area.
Thus De Moivre under Newton's Tutelage developed the complex polynomial arithmetics based on the figure of the unit circle. To this figure much is owed. Newton used it to develop the Differential calculus for the binomial series, and the calculation of fluxions. moving dynamic forms especially orbits. De moivre used it to develop the de Moivre Cotes theorem that linked polynomial solutions of the conics to the Trigonometric ratios, and also to the Logarithmic tables.
The De Moivre Cotes theorems are a remarkable set of theorems in the complex number field which link trigonometric ratios to complex numbers and logarithms. In fact Cotes was able to come up with the value for e to a certain number f places, a feat which was not his alone, as both he and Newton enjoyed a game of calculating it. It was Cotes who described the logarithmic version of the Cotes Euler identity about 70 years before Euler established the exponential version, using Transcendental formulae like Newton's Binomial series expansion.
Cotes Died shortly after this and Demoivre who had been collaborating with him on certain subjects using these principples finished off a piece of work for Cotes before moving into his are of new interest , an application of the very same principled to what has now come to be called Probability.
Thus De moivre used the link between the trigonometric ratios, logarithms and the unit circle to completely define and describe the measurement of probability. Towards the end of his work he established the exponential version of probability distributions, and utilised certain hyperbolic sine and cosine formulations of it. This was nearly 2 centuries efore the development of Quantum mechanics as a field of study.
The application of measurement to the many and varied siuations in space is a philosophical and a metaphysical problem which rightly does not belong to mathematics, but to metaphysics. But since such subject boundarues are subjective, fluid and subject to revision and redaction, in short in confusion, The growing identification of a core of studies called mathematics with the determination of the concept of number inevitably fused the two together in the popular mind.
I do not need to rehearse my research or my opinion with regard to the concept of number but sudffice it to say that at a time when thinking people were struggling to ind a form of consistent trustworthy expression of complex relationships, the high Platonic opinion of mathematics placed it in a unique position of trust. Yet Plato meant different things when he opined about mathematic, than men in the early twentieh century received of him! Thus plato admired the ordere scientific philosophic and locical analysis found in the Gematrial work of the Astrologers/astronomers of his day, which by his day had come to be called mathematikos, bein ordered scientific thinking, especially about the motion of the stars and planets. Thus the "gods" of ancient greece were well described and worshipped uniquely through mathematics.
This veneration of the Gods of nature came down to the twentieth century idea that nature can only truly be described by mathematics! This is of course patent nonsense, but it was and still is a mystical belief that some hold about this new collection of ideas which grew up over time to replace what was calle mathematics in the past, that is scientific thinking.
Thus we find in the brave and adventurous 18th t0 19th century, that many chancing men boldly went wher none had gone before, developing the great innovations of the industrial revolution, the great technological and scientific advances using the scientific method, and the apprehension of nature and its element in the development of the atomic theory of
All these advances rested on a singular belief that man was equal to the task of ruling the planet by divine institution and spiritual impetus, and many insightful "breakthroughs ", hailed as divine inspirations, were in fact ancient Greek and Indian and Arabic notions given new life by the industrial revolution and the profit motive. It did require an industrialised sector of the world to drive thes innovations, because otherwise there was no incentive or market or empire to return the capital investment to the entrepreneur with profits.
Consequently, starting with the dissolving of the holy roman empire, and Prussian response to the increasing industrialisation of the western nations of Europe, and the burgeoning wealth of the newly colonised Americas, there was little truck held for esoteric philosophising and splitting of hairs. That time had gone since Kant essentially removed the basis for division in the rationalist camp. Now pragmatists in the form of industrialists who could see great wealth beckoning only wanted technology that worked and turned a profit, regardless of what metaphysics or philosophical laws applied or did not apply.
Academia struggled to remain relevant in this zeitgeist, especially when successful entrepreneurs gained social mobility and upset the social order and class system, and even the academic system based on it. Now som artisan who had made a fortune could opine on the nature of things and tha causes of reality, and be listened to because of his success and in spite of his lack of learning. In fact learning became a commodity bought by these wealthy individuals or corporations whenever they neede it!
Part of what drove this phenomenal success was the analogising of centuries old knowledge to apply in related but not previously connected fields. Thus dropping back to the mid to late 19th century many in academia were waiting for a system of measurement to make it possible to cope with the calculation of data derived from empirical fields like electricity and magnetism. The analogy was Newton, who had successfully used data to develop a theory of gravity, but the theory of elecric fields and Magnetic fields required a different type of mathematical measurement structure, it seemed to require more than 3 dimensions. IN this case the notion of dimensions was well understood as parametric measuring tools, not weird notions of space.
The Advent of the just in time quaternions was reat scientific event which seemed to offer a solution. This was a new systematic apprehension of space using these 4 dimensions discovered by Sir William Rowan Hamilton. To Hamilton they were so significant that he set out to rewrite the Euclidean manual Called the Element based on them. He felt that they were that significant, and we could "no longer think in terms of 3 dimensions only", as if we did!
However, despite his extraordinary intellectual tour de force, there wer others who were not so persuaded, and in particular favoured a different , piecemeal approach which gradually developed, borrowing terms and ideas from all the advances occuring at the time and shaping itself into a subject area that Gibbs defined as Vector Algebra.
Vectr Algebra was a developmental strand of a more fundamental analytical system devised by Hermann Grassmann. One thar Hamilton acknowledged as being superior to his own current work, but aimed differently! And this is the point Hamilton, had found wht he thought were an analogous number system called quaternions. He was sucked in by the idea that the grear achievement of mathematics was the development of the number systems, whereas grassmann wasnot even thinking mathematically he was thinking about the philosophical approach to knowledge acquisition that was hermeneutic and applicable acros all areas of Wissenschaften. It is only in !862 when Grassmann is forced to demonstrate the power of his analytical system in mathematical terms that Gibbs and others realised they could construct their own 4 dimensional systems to order, not having to rely od discovery or divine providence. Consequently Hamilton's Quaternions lost their divine force in America after Gibbs championed the new ideas of Grassmann as his own. Also Hamilton was atacked at home by none other than Lewis Carol. Lord Kelvin and Maxwell who had been at first an ardent supporter of Quaternions. Quaternions were thus quite quickly back watered in favour of the new Gibbsian vectors, and the matrix and determinant theory that followed on from them, with the later Tensor theory. All of hese in fact were in some way developed by Hamilton but nobody trusted quaternions after that spin doctor campaign against them.
Mean while, in eastern Europe, while this was going on in the west, Philosophers there decided that the nature of the electromagnetic field was such that a new class of "numbers were needed to describe them. Einstein had moved to the west Bringing this eastern philosophy and connections into the heart of western physics, but Minkowsi and others in the east developed under their own cultural and philosophical interests, and they began to recognise the potential for the complex numbers, now definitely accepted as numbers.
Schroedinger and others realised the connection between all these things estblished by De Moivre Cotes and Newton. Thus he was able to compltely recast wave theory in terms of the complex numbers And through De Moivres link to probability he was able to interpret the complex valued wave equation in terms of a probability distribution.
AS odd as this sounds or reads, this analogy is purely formal. The power of it is that instead of thinking in a disjointed way about bits and pieces that need to be considered at the same "time" the complex numbers enabled the load that had to be remembere to be substantially reduced in the combinatorics, enabling necessary pieces to be isolated only when necessary. Thus the complex numbers really say nothing about space, they merely simplify manipulation.
However, later, when complex numbers were identified as vectors, then they began to be interpreted as making some statement about space. Minkowski also realised that to describe the behaviour of space time he had a formal relationship between the complex number and the rotations in spacetime.
Thus for many good reasons complex numbers became important in writing and manipulating the equations involved in theoretical physics, and the link to probability was and is a formal link, but it has found an explanatory value in the Heisenberg uncertainty /certainty principles.
When Einstein decided to use Ricci and Lucas Tensor theory to describe the systems he was manipulating every day, and which, even with complex numbers were unwieldly, it was precisely becauseit eased the combinatorial load. Nobody seriously says that space is some kind of tensor number system, whereas many still elieve that space is some kind of quaternion number system even if it is made up of pairs of complex numbers in a Clifford algebra.
What we see here is the hypocrisy of mathemaicians, who at times want us to belive these numbers have spatial and geometrical significance, rather than formal metrical nd combinatorial cost benefits, while at other times they uses a system of tensors for the same reason, but do not want to add them to the number concept.
The number line concept has been supported by ring theory and group theory descriptions of it that preclude and exclude certain collections of symbols from the "hallowed name", number. While formally this distinction is significant, scientifically it is not. The application of numbers to scientific measurement says nothing about he space to which they are applied but much about us who apply them. We are lazy and feeble and prone to mistakes, thus anything that saves work, saves our mental energies , and cut down on error we are going to use precisely for those reasons.
To attribute some inner truth to the symbol we use is consistent with the history of Pythagorean Theurgy, but it is our subjective process to which the meaning of the symbols devolve, not to the space which we also subjectively process in some analogous way.
So for example. rotatio occurs in space, this can only be modeled if we have a representation of space as atool whivh records this rotational motion. If i use complex or quaternion numbers to record this rotation, it is perforce by relationship to this representational system, not the quaternions themselves.Thus there are many other ways to record this rotation that may be better or worse than Quaternions. The numbers have no special significance .