[I completely rewrote the question to see if I could make it clearer. The comments below won't make any sense. In fact, my original question has been answered by Eric Wolfsey, so I may restore it.]
When you read about the quaternions on Wikipedia and on many other sources, they are defined with the relation $$i^2 = j^2 = k^2 = ijk = -1$$ This comes across as completely random. There is no explanation of where this relation comes from.
These sources then go on to prove that the quaternions satisfy various algebraic and geometric properties. Among these properties, they act on 3D vectors as rotations (but as a double-covering), they act on 4D space as rotations, they're associative, they're distributive. Etc. All of this comes across as a random coincidence when you compare it to the defining relation.
Some of the time quaternions are thought of as 3D rotations. But this is a lossy interpretation, as a quaternion $q$ and its negation $-q$ represent the same rotation. For similar reasons, addition of quaternions starts to seem more bizarre.
When defined as 4D rotations, they happen to be an "isoclinic rotation". I haven't thought enough about this concept...
When defined as a complex matrix that satisfies a linear algebra relationship (sourced from here: https://qchu.wordpress.com/2011/02/12/su2-and-the-quaternions/), closure under multiplication is baked in, but closure under addition comes across as a coincidence. This relation is: $$M\text{ is a $2\times2$ matrix over $\mathbb C$}, M^\dagger M \in \mathbb R, \det(M) \geq 0 $$
Geometric Algebra textbooks show that they're an instance of a larger family of algebras with similar interpretations, called Geometric Algebra (or Clifford Algebras). But then why do Clifford Algebras exist? Why do they have their mixture of algebraic and geometric properties? It feels like you're replacing a small mystery with an even bigger mystery.