Quaternions? Basis of quantum mechanics? Pretty sure that's not right at all. A lot of games use them for rotations in place of rotation matrices though I suppose.

Quaternions are not the basis for quantum mechanics. Biquaternions have some applications in quantum field theory, but there are many areas of quantum mechanics where there's no need or space for anything above complex.

We owe "quatties" to hamilton, but there is a generalization of the process in case you're curious: https://en.wikipedia.org/wiki/Cayley%E2%80%93Dickson_construction

The general concept is called Spinors, Quaternions are just one representation. Here's a great video on them. In physics they're using them because they're necessary (video explains), in computer graphics we're using them because they're algorithmically convenient, very cheap to compute and ignore that whole half-spin thing. It's one of those instances where it's cheaper to compute useless information and then throw it away as opposed to avoiding to compute it.

They're also absolutely impossible to deal with when authoring stuff, as in rotating things in Blender, it's just a representation on the backend. Quaternions would avoid gimbal lock but when authoring you really rather deal with that than a 4-dimensional hypersphere.