You don't need the full theory of complex numbers. Start with a zero vector, then iterate doubling the angle, squaring the magnitude, and translating by the vector you're evaluating. It's not totally inconceivable that somebody could have come up with it by just messing around geometrically.

But yes, good instincts.

Wow, thanks! Never heard the geometric explanation and it, as usual, is way more intuitive.

Consider this formulation of complex numbers:

Unit is 2x2 identity matrix.

imaginary unit is the matrix

    -1 0
     0 1

It is easy to check that imaginary unit squared is -1. And addition and multiplication is just matrices.

It then quickly becomes apparent that all possible matrices you generate _are_ the rotation and scaling matrices. Instead of drawing a connection between rotation and complex numbers, they all of a sudden are the same thing.

surely it should be:

    0 -1
    1  0

instead?

Wholly correct, sadly I cannot edit anymore :(