I think the issue is that quaternions have just too many interesting properties. And everybody is interested in just one aspect, so for one they “are” a convenient way to work with geometric calculations (game programmers), for others, they “are” a generalization of a certain class of linear operations (linear algebra guys), for others, they “are” the next generalization step beyond imaginary numbers (analytic math). And all are right, but every text you get is highlighting one of the aspects and completely ignoring the others, so you get confused since all the descriptions seem to have nothing in common.
At least that’s the understanding I have arrived at. Which is a bit of “at least I know why I don’t understand them” I’d say
The main misunderstanding I see about unit quaternions is their true role in the avoidance of the gimbal lock. Tons of people don’t know that if you use them with eulerian transforms, they won’t solve this problem.
I worked a bit on non eulerian transforms at the end of my License degree. The only other way to fix this problem (gimbal lock) but in a non linear way (without unit quaternions) consists in taking into account the singularities on poles.