The "imaginary" part of the wavefunction is no less, or more, real
than the "real" part. The wavefunction is simply a two-component
object, and I think it's somewhat unfortunate that we attach the
names "real" and "imaginary", with all their unwanted connotations,
to the two components.
Why does the wavefunction have to be a two-component object? Because
a single component could contain all the needed information about
the position of a particle but not also the momentum (or vice-versa).
If the wavefunction is to encode a *complete* description of the
state of a particle, then one component simply isn't enough.
For example, try to find a way to use a single-component function
to describe a particle with a well-defined momentum (including
direction) and completely undefined velocity (equal chance of
finding it at any x).
Given that the wavefunction has to be a two-component object, why
is it so convenient to use complex numbers to encode the two
components? Beats me. Always seemed a little spooky, to tell
the truth.