This is where my answer is a bit messy. Imaginary numbers can be thought of that way, but using them for 2d space is an application.

We do have quaternions. This gives us 3 imaginary numbers, we call i, j and k, where squaring any of them gives -1, and ij = k, jk = i and ki =j. Weirdly, we lose commutivity here. ij = k, but ji = -k. Reversing the order gives a negative result.