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.