By Putting a context around Hamilton much can be gleaned about his motivations and predilections. Chiefly his religious and scientific persuasions placed him in a fierce running stream of innovation and innovative thinkers. Grassman, Rodrigues, et al were all pursuing the same goal he was, a goal first toyed with by Gauss: a way of compounding in 3 coordinates, the motion of objects, particularly rotations in many axes. The impetus was astronomy, but also dynamical systems were demanding a coherent approach.
There is no doubt that Hamilton was widely connected, and also his work was known, but it was his work on couples that propelled him into the limelight. He revolutionised the conception of imaginary numbers and put them on a sound footing as complex numbers. This is not to say that they were not in use prior to him, they were, nd with an extraordinary confidence that merely enhances the intellectuality of Newton, De Moivre, Cotes in particular. Euler and was a little tentative about them, but Gauss seized them to demonstrate the fundamental theorem of polynomials. As such Gauss understood, again tentatively that the "numbers" had something to do with geometry of space. Much of his thinking, however he kept private, because i believe he was Aspberger's Autistic, especially as He got older.
he gave Riemann such a dismissive response that it is hard to explain his behaviour in any other terms.
Curiously Hamilton began his development of Couples by utilising time, but as i found out this was a "con", a notion based securely on ordinal numbers. I must say he owns up to it as a necessary step to advance his exposition, and i daresay in his time it would have been understood as a brilliant abstraction of the underlying notions in ordinals, and a dazzling verification of the secure validity of "ordered pairs of numbers".
To go from ordinals to ordered pairs of numbers naturally is some feat, but this is achieved by anchoring the notion in time flow, and then somewhat hypnotizing the reader into excepting the notion there, and then immediately linking this to the ordinal numbers and immediately to the numerals, thus giving the reader a secure mental root to the concept of ordered pairs of numbers, and relating it to the flow of time.
Hamilton by generalising his notion can extend it to the broad expanse of a flowing river of tim and thus to the moving plane, in which he naturally takes instances. This such as it is is the basis of the complex plane.
The beauty of the number he has pulled on the reader is that it is exactly what is needed to switch the dullard mind from the confusion of "imaginary" numbers to the realisation that the geometrical plane is what is being discussed and referenced, but not by numbers. Rather by ordered "measurements" , that is flows of measuring actions. These he called vectors. He also introduced the trigonometric ratios into the subject, late in the day to encompass all that Wesler had done and Argand and others , and the work of Napier. This last reference was in support of his dear friend who had been disgracefully vilified by some "upstart" to the royal society of Scientists, for a theorem using imaginary logarithms.
It was and is a tour de force, one which he hoped to quickly follow by the theory of triples. This proved intractable.
In the meanime growing voices in the electric magnetic communities were searching for a simpler, coherent way to represent the geometric properties of light , electricity and magnetism. Hamilton, a renowned synthetic geometer had caused no smal stir in his doctoral thesis by re constructing certain elements of the theory of optics in a more algebraic way, and correcting some past errors in the topic. So when he saw the need and indeed the progress and striving of others to fulfill the need, he clearly woke up from his arduous task of getting the triples to work for rotation in 3d space and attempted a quaternion set. Theoretical Physicists and Mathematician par excellence James Clerk Maxwell was looking for an algebra of 4, as were other theoreticians. Gauss had intimated but not explained how it would work. It seemed fantastical then, but now Maxwell realised that the laws of light and Electricity and magnetism were so similar that a 4 algebra was needed to account for the differences.
Hamilton obliged. His 4 algebra not only contained the complex numbers and thus the reals as an underlying ordinal set, but also enabled 3d rotations to be described. He immediately applied it to spherical trigonometry, because this meant they had astronomical significance, and thus he could specify te 4 algebra for many real situations. One in particular caught his attention:time plus space.
Maxwell immediately seized upon the work to express his notions, and this from the outset gave Hamiltons Quaternions an extraordinary significance. Hamilton never recovered from the blinding significance of his "discovery" and deoted he res of his life to it,
Now were we to stop there we would perhaps feel justified in being shocked at the way Hamilton ws treated toward the end of his life: the pot of gold had turned to ashes.
Hamilton's Quaternions ruled the day, but under its wings Grassmans vector algebra became comprehensible. Thus it was found that Grassman had an alternative exposition of the subject and that Rodrigues had still another exposition. The question was which one was going to win the day?
With Maxwell on board it seemed assured that quaternions would carry the day. However, Maxwells initial enthusiasm wained, particularly because o the difficulty of the subject in terms of the number of terms a calculation produced, followed by the non commutative nature of tge actions. Like Euler, Maxwell was prone to make simple mistakes of habit, due to the fact that we are schooled in commutative actions, this was annoying and irksome because he had to go through he whole calculations term by term to see where he had made the mistake!
In the meantime commutative forms of vector algebras were comming on stream, mostly derived from Hamilton's Quatrnions, and Maxwell was more inclined to se thoe than Quaternions as time went on. Each bauldlerizer of Hamilton's work set their work up on distinct competition to his! Hamilton had defined a couple of products : the dot product and the scalar product. Someone else defined the crossproduct based on Hamilton'work. The nabula became defines and the curl. Each of these vector operators was commutative. Eventually Maxwell shifted completely to these vector forms and the American Academy of science decided against Quaternions. Hamilton died a shattered man. However this was an emotional response to a critical time in his life.
I do not know if Hamilton hoped to make finacial gain out of his discovery, in which case his hopes were dashed, but as far as his invention was concerned it had survived transformed by and transmuting its vector basis . Today Doug Sweetser can write Quaternions in vector notation and do the fundamental manipulations in a clear and transparent way. When it comes down to it, that was the main beef that Maxwell had about Quaternions: they were not transparent. Deeply significant, essentially indicative, but horribly messy. Nobody could read Grassman because he was so terse. It took years to translate his terse notation into transparent expressions. Similarly Quaternions were so voluminous that it was overwhelming. It took years and a whole team of reformulators to prune it down to a manageable system of notation. You can enjoy it today at
Ironically. a few years after his death William Kingdom Clifford generalised Hamilton's Quaternions using a Quadratic valuation as opposed to a linear valuation of the product of ordered set. This was found to be even more applicable than quaternions, and especially in Electromagnetism etc. Levi and Ricci followed this up with Tensor structures and it became clear that the properties of physical space required as many independent variables as one could manage!
Laplace and Lagrange, Bernoulli and Stokes-Navier all attested to this, and so quaternions were destined to become, like Newton's Physics a foundational milestone along the way to a more complex description of physical space. The beauty of Hamilton's contribution is that it literally smoothed and paved the way for modern Physics, and many of Hamilton's notions and notations are indispensable. He may yet have notions and ideas as yet unmined among his papers, but clearly Clifford has superseded his preeminence, algebraically, but it i easy to see how to supersede Clifford Algebraically and topologically. The point is :applicability.
Today Quaternions remain as applicable as Newton's Principaea Mathematica, and are more accessible than the general Clifford Algebras, just like Neton Mechanics is more accessible than Einstein's.