Where derivation == "intuitive routes", since a "law of physics" cannot be derived, only observed right or wrong.
Some approaches:
Video 1. Necessity of complex numbers in the Schrödinger equation by Barton Zwiebach (2017) Source.
This useless video doesn't really explain anything, he just says "it's needed because the equation has an in it".
The real explanation is: they are not needed, they just allow us to write the equation in a shorter form, which is always a win: physics.stackexchange.com/questions/32422/qm-without-complex-numbers/557600#557600