"Any system with this mix of properties, commutative or not, is called a division algebra. The real numbers and the complex numbers are division algebras, be- cause we don't rule out commutativity of multiplication, we just don't demand it. Every field is a division algebra. But some division algebras are not fields, and the first to be discovered was the quaternions. In 1898, Adolf Hurwitz proved that the system of quaternions is also unique. The quaternions are the only finite-dimensional division algebra that contains the real numbers and is not equal either to the real numbers or the complex numbers." (Ian Stewart, "Why Beauty Is Truth", 2007)
"Quaternions have developed a strange habit of turning up in the most unlikely places. One reason is that they are unique. They can be characterized by a few reasonable, relatively simple properties-a selection of the 'laws of arithmetic', omitting only one important law-and they constitute the only mathematical system with that list of properties." (Ian Stewart, "Why Beauty Is Truth", 2007)
"The quaternions arise when we try to extend the complex numbers, increasing the dimension (while keeping it finite) and retaining as many of the laws of algebra as possible. The laws we want to keep are all the usual properties of addition and subtraction, most of the properties of multiplication, and the possibility of dividing by anything other than zero. The sacrifice this time is more serious; it is what caused Hamilton so much heartache. You have to abandon the commutative law of multiplication. You just have to accept that as a brutal fact, and move on. When you get used to it, you wonder why you ever expected the commutative law to hold in any case, and start to think it a minor miracle that it holds for the complex numbers." (Ian Stewart, "Why Beauty Is Truth", 2007)
"The system of quaternions contains a copy of the complex numbers, the quaternions of the form x + iy. Hamilton's formulas show that -1 does not have just two square roots, i and -i. It also has j, -j, k, and -k. In fact there are infinitely many different square roots of minus one in the quaternion system." (Ian Stewart, "Why Beauty Is Truth", 2007)
"However the nature of mathematics itself has led us, at first reluctantly, to go beyond real numbers to the realm of the so-called imaginary and complex numbers. Moreover modern mathematicians also deal in infinite numbers of more than one kind, and also quaternions, octonians, and matrices, which can be regarded as another generalization of number." (Peter M Higgins, "Number Story: From Counting to Cryptography", 2008)
"Quantum theory may be formulated using Hilbert spaces over any of the three associative normed division algebras: the real numbers, the complex numbers and the quaternions. Indeed, these three choices appear naturally in a number of axiomatic approaches. However, there are internal problems with real or quaternionic quantum theory. Here we argue that these problems can be resolved if we treat real, complex and quaternionic quantum theory as part of a unified structure. Dyson called this structure the ‘three-fold way’ […] This three-fold classification sheds light on the physics of time reversal symmetry, and it already plays an important role in particle physics." (John C Baez, "Division Algebras and Quantum Theory", 2011)
"Quaternions are not actual extensions of imaginary numbers, and they are not taking complex numbers into a multi-dimensional space on their own. Quaternion units are instances of some number-like object type, identified collectively, but they are not numbers (be it real or imaginary). In other words, they form a closed, internally consistent set of object instances; they can of course be plotted visually on a multi-dimensional space but this only is a visualization within their own definition." (Huseyin Ozel, "Redefining Imaginary and Complex Numbers, Defining Imaginary and Complex Objects", 2018)
"[...] quaternions give a nice approach to symmetric objects in three-dimensional space: the regular polyhedra. But this leads in turn to the regular polytopes, a family of four-dimensional symmetrical objects as remarkable as the regular polyhedra. One then becomes convinced that four-dimensional space is not just a set of quadruples; it is a world of genuine geometry." (John Stillwell, "Yearning for the Impossible: The Surprising Truths of Mathematics" 2nd Ed., 2018)
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