> A quaternion is basically a 4 dimensional vector, so it has a magnitude (or norm, or length)

Is it really a vector in the physical sense? People often say vector when they mean N-tuple -- for example we learned in high school that vectors are just N numbers taken together.

For physicists, a vector must satisfy certain transformation laws - it must transform in the correct way if a rotation is applied, and the scalar product must be invariant of the coordinate system, IIRC. I don't have enough intuition of quaternions to say how they behave under transformations, though. I would be surprized if you could have "proper" vectors with four components in three-dimensional space.