>it does anything mathematically interesting

You are right - its not interesting. You already know that rotation can be done through multiplication (i.e rotation matrix), and you are just simplifying it further.

After all, the only application of imaginary numbers outside their definition is roots of a polynomial. And if you think of rotation+scaling as simple movement through the complex plane to get back to the real one, it makes perfect sense.

You can apply this principle generically as well. Say you have an operation on some ordered set S that produces elements in a smaller subset of S called S' It then follows that the inverse operation of elements of the complement of S' with respect to the original set S is undefined.

But you can create a system where you enhance the dimension of the original set with another set, giving the definition of that inverse operation for compliment of S'. And if that extra set also has ordering, then you are by definition doing something analogous to rotation+scaling.