I didn't think quaternions are even exotic and have used the extensively to represent rotations. I'm not familiar with video games but it seems a pretty natural fit to the problem. What is the alternative that is so much better?
I just dabbled in webgl/threejs and tried creating a small movement engine and was confused from beginning to end. Quaternions as a black box is pretty accurate.
BTW, it's easy to understand the relations between i,j,k if you're familiar with Pauli matrices.
I propose we call them fanciful numbers.
The fact that it’s a “quaternion” or a “rotor” is kind of an implementation detail.
Of these four terms, Quaternion is the most precisely correct. The reason is that rotors are, by definition, constrained. Quaternions can take any value. Due to floating-point precision problems, your rotor will not always be exactly a rotor, but may some multivector which is not a rotor.
This is kind of like representing a point on a sphere using (x,y,z) coordinates. You can call it PointOnSphere or Vector. I would rather call it Vector, because PointOnSphere implies a constraint which won’t be exactly satisfied, and I want to be reminded of that fact. The type name (Vector here, and Quaternion above) represents the object’s structure and the constraints which are actually enforced by the underlying representation.
This is unsurprising when several of your coordinates become -1 when multiplied. That's why historically Gibbs Heaviside (dot and cross product) became the dominant vector algebra over quaternions.
Clifford Algebra is the better than both, as you can seamlessly do dot, cross (wedge in CA) and can also embed quaternions within the system. I've heard that it can also accommodate some of the nonmetrical aspects that make differential forms appealing for manifold integration, but that's currently outside of my range of knowledge.