Only if you’re describing orientation as two orthogonal rotations. I’m saying think of it like a ‘pointing’ vector that defines the axis of rotation. And such a vector does require 3 components in 3d space
> Only if you’re describing orientation as two orthogonal rotations.
No, the space has the dimensionality it has. You may choose to describe a 3D orientation with more than two numbers, but you won't stop it from being a two-dimensional quantity that way. If you use more than two numbers, those numbers will fail to be independent of each other.
Your comment is essentially correct if you replace the word "orientation" with "direction", though.
But I agree it is helpful to think of quaternions as direction and spin.