Imho, the presence of i in the schrodinger equation has a simple origin. The equation describes evolution of a wave and waves can be conveniently described with

Imho, the presence of i in the schrodinger equation has a simple origin. The equation describes evolution of a wave and waves can be conveniently described with amplitude and phase components bundled together. With such description a wave can be moved or shifted by multiplying it by a complex number. Otoh, if the same wave is described by real numbers only, each number meaning the displacement from the null state, then shifting the wave pattern becomes a non trivial operation.