[stared]

In short:

scaling -> real numbers

1d rotations and scaling -> complex numbers

2d rotations and scaling -> quaternions

In the case of quaternions, there is called double-covering, which turns out (rather than being an artefact), play fundamental role in particle physics.

[conformist]

Yeah sure SU(2) up to sign is isomorphic to SO(3) and whatnot… I think it’s probably mostly the computer graphics history and the cool name that gets people excited about quaternions?