I can deal with the idea of amplitude have two dimensions and rotating through them, and with amplitudes canceling each other out because of this, but it's not clear to me, if there is "no preferred direction" for amplitudes, why the square root of minus one is involved. Are there other mathematical formulations possible?

Matrices of the form ((a, -b), (b, a))---that is, compositions of a scaling and a rotation in the plane---are isomorphic to complex numbers, which makes sense because when we multiply complex numbers in polar form, we multiply their magnitudes and add their arguments.