The Quaternions are the smallest.... I've been reading https://github.com/GleasSpty/MATH-104-----Introduction-to-Analysis, and the author formulates the integers as the smallest (by inclusion under isomorphism) nontrivial totally ordered cring that contains the natural numbers, the rationals as the smallest totally ordered field that contains the integers, and the reals as the smallest dedekind-complete (or cauchy-complete) totally ordered field that contains the rationals. Similarly, there's the algebraic numbers which are the smallest (edit: they're not totally ordered) algebraically complete field that contains the rationals, and the complex numbers which are both algebraically complete and dedekind-complete.
Is there a similar statement for the Quaternions/Octonions?
 A: By Frobenius' theorem, the quaternions $\Bbb{H}$ can be characterized as the smallest noncommutative division ring that contains $\Bbb{C}$.
A: If I understand correctly what you are looking for, then no, I do not believe there is a comparable statement to be made about the Quaternions and Octonions.
Quaternions and Octonions are useful, but I would not say that they are remarkable because of their structure in the same way that Integers, Rationals, Reals, Algebraic Numbers, or Complex Numbers are. Quaternions are not commuatative, and Octonians are neither commutative nor associative. So rather than having more structure than any other set of numbers, they actually have less.
Quaternions are specifically useful because they can represent mathematical objects (i.e. 3D rotations) in a way which makes them easy to manipulate and reason about. It is easy to compose and interpolate rotations when they are represented as Quaternions because these operations are "encoded" in the algebra of the Quaternions.
It is my understanding that Octonions also represent some algebraic objects more elegantly than other sets of numbers do, and can be used to practical ends for that reason, but I am not familiar with how they are used.
