He's talking from a mathematical perspective, not a compute graphics one.

(Unit) quaternions happens to work as rotations in 3D space. But quaternions' algebraic structure is indeed 4D, just like imaginary numbers' is 2D.

Since you can multiply a quaternion by a quaternion, the quaternions act on themselves, so they indeed act on a 4D space. They can also act on a couple 3D spaces: the unit sphere in 4D, called the three sphere, or the imaginary quaternions.

In math, this is called a representation of a Lie group, and there are representations in all dimensions.