Plotting imaginary numbers on a separate axis from real numbers to create a plane is an obvious idea. Got another dimension, add another axis, right?

The rotation pops out from the binomial multiplication of real + imaginary pairs: (a + ib) (c + id).

And the way that is done is nothing new: it follows the same old FOIL rule: ac + iad + ibc + i^2bd. Where i^2 gives us -1 so we get ac - bd + i(ad + bc).

When you plot a few points on the complex plane and try multiplying them with the above, you will soon discover that their arguments (i.e. angles) are adding together, and you can then prove that with some basic trig.

For instance if you multiply together any points that are both 30 degrees off the real axis, you get a point that is 60 degrees, and so on.

You can completely removce the "i" from the picture and just have the multiplication rule as <a, b> x <c, d> = <ac - bc, ad + bc>, and then see what happens when you convert to polar coordinates.

You can connect that to linear agebra, in which rotation of a vector is achieved by multiplication by a suitable matrix (and in connection to the above, such a rotation matrix will have some complex eigenvalues; real ones correspond to scaling/shearing).