What does real and imaginary part of wavefunction signify?

Question

Does real and imaginary part of wavefunction carries any physical interpretation?

Answer 1

A complex number $x+yi$ can be written as $re^{i\theta}$. The real and imaginary parts do not matter as much as the modulus $r\geq 0$ and phase $\theta$.

The phase factor does not on its own affect probabilities of observing something observable, since they are calculated as $|z|^2=r^2$, but when adding wavefunctions it can interfere in ways that are observable (e.g. two electron orbitals with opposite phase will produce a sum of zero, indicating that the electrons repulse each other).

As Qmechanic pointed out, physics is unchanged if everything is multiplied by an overall phase $e^{i\theta}$. So there is nothing to distinguish our standard real-imaginary coordinate system from any rotated coordinate system.