Because in 1985 Ken Shoemake dropped the idea like a bomb on the computer graphics industry and it changed the way hackers thought about rotations forever. https://www.ljll.fr/~frey/papers/scientific%20visualisation/...
I mean, there are practical reasons too (which are mostly just isomorphic to the stuff in the paper). But really that's why. It's part of our cultural history in ways that more esoteric math isn't.
Thank you for the reference, this is one of the more approachable sources for learning the nuts and bolts of quaternions that I've seen!
Because people like me use quaternions but have never attained a full understanding like 3x3 rotation matricies. I will be reading the above link since its only 12 pages and someone indicated its an easier read.