I would think that the advantages are conceptual. I never learned quaternions on their own (unless I've repressed those memories), but I was aware of them as I

I would think that the advantages are conceptual. I never learned quaternions on their own (unless I've repressed those memories), but I was aware of them as I was learning geometric algebra. My experience with learning complex numbers in an undergraduate electrical engineering class was that these were baffling mathematical objects that make the equations work, but understanding why or how was best left to the mathematicians. Names like "complex" and "imaginary" didn't help. I would imagine that learning quaternions on their own would be similar. Quaternions are like complex number, but more so, and they have this i*j=k property that you just have to accept as part of how they operate. So you have this four-dimensional object that performs rotation in three dimensions, and that's just how it is. The geometric algebra construction explains all of these mysterious properties and gives an intuitive algebraic and geometric sense for why they behave that way. And this intuition extends beyond quaternions. Projective, conformal, and spacetime geometric algebras all have equivalent objects that behave differently but according to the same fundamental rules.