TITLE: Why should quaternions exist?
QUESTION [35 upvotes]: Why do quaternions exist?
I want to believe they exist, but all I can think of are reasons they should not exist.
These are my reasons.

The quaternions are defined by the following equation:

$$i^2 = j^2 = k^2 = ijk = -1$$
There are four equalities and three variables. There should be no solution.

Somehow there's a workaround for dilemma $1$. They exist.
So how do you define the quantity $ijkijk$?
Is it interpreted as $(ijk)(ijk)$, or $-1 \times -1$, or $1$?
Or is it interpreted as $(i^2)(j^2)(k^2)$, or $-1 \times -1 \times -1$, or $-1$?
$ijkijk$ can't be two values at once.
I read that quaternions are not multiplication commutative. Why multiplication commutative? Why not multiplication associative or some other property?
Okay, so there is a reason for number 3.
Why can't I define a new set of numbers where the multiplicative identity doesn't hold?
Like: $1x = x + 1?$

I've been pondering these questions for a couple of days now, so I would really appreciate an answer.

REPLY [2 votes]: The quaternions are an example of a finite-dimensional associative real division algebra. Frobenius proved that there are only three: the reals, the complex numbers, and the quaternions (the octonions are nonassociative). So if you are inclined to believe division algebras exist, then the quaternions probably do.
On the other hand, it is a consequence of Bott's demonstration of a conjecture by Hirzebruch on the Pontrjagin classes of vector bundles over n-spheres that the sphere $S^{n-1}$ is parallelizable only for $n-1$ equal to 1, 3, or 7. So if you believe 3-spheres exist and that you can parallelize them (put an everywhere full-rank smooth vector field on them), then you should believe that $\mathbf R^4$ possesses a bilinear product operation
without zero divisors, and hence that the quaternions exist. But you don't have to.