One of the big motivations for quaternions is performance. Not only are they much more compact than a rotation matrix (4 float vs 9), but many important operations, such as composing rotations, are also much quicker. The only real downside of them is that they seem to drive everybody to this obsession to try to truly understand them. After years of using them, I've learned to embrace my ignorance. The underlying math is not terribly complex (hur hur), but relationships between interdependent periodic rotations all packed in a tidy imaginary number package, is simply not something you're going to really get a good intuitive feel for - numerically speaking.