TL;DR

In Schrödinger's original papers, he introduces an ad hoc definition of the wave function (sometimes called an ansatz) that leads to his famous equation. The definition was ψ = exp(S/K), where S is the action. However, with K real, as the definition suggests, it did not work; but when K = - i*(hbar) it did. Here, how this was corrected and why the definition works is examined. It also explains why QM needs complex numbers.

For those interested in how Schrodinger originally derived his famous equation and why it eventually gave the correct answer (after some meandering), see:
https://arxiv.org/pdf/1204.0653

Note (with some minor paraphrasing for clarity):
'Schrodinger’s ansatz is just the fundamental second postulate of Feynman’s formulation of quantum mechanics for the time dependence of a path amplitude. It was, therefore, in a sense, not necessary for Feynman to demonstrate that the Schrodinger equation follows from his postulates, since, in fact, Schrodinger had made, a priori, essentially the same postulate in order to derive the equation.'

It could be developed into an interesting introduction to QM after a classical mechanics course like Morin:
https://www.amazon.com.au/Introduction-Classical-Mechanics-Problems-Solutions/dp/0521876222

Thanks
Bill

Let's be honest here. Feynman's work is an extension of Dirac's, so we only need to go to page 68 of Dirac's 1932 article to find what we want. Sure, Feynman "had a programme", and he literally "went back to Schrödinger", but putting i times Lagrangian action had been done by Dirac.