On the "reality" of the wavefunction, there seem to be two schools of thought on why treating $\psi$ as something more than a mathematical tool is erroneous:
1. $\psi$ involves complex numbers. Only Real numbers correspond to measurable quantities.
Who thinks that? Complex numbers are usually associated with a very real rotation. As for the reality of the wavefunction, take a look at weak measurement work by Aephraim Steinberg et al and by Jeff Lundeen et al:
"Direct Measurement of the Wavefunction
Chosen as 2nd most important Physics Breakthrough of 2011 by Physics World!
Central to quantum theory, the wavefunction is a complex distribution associated with a quantum system. Despite its fundamental role, it is typically introduced as an abstract element of the theory with no explicit definition. Rather, physicists come to a working understanding of it through its use to calculate measurement outcome probabilities through the Born Rule. Tomographic methods can reconstruct the wavefunction from measured probabilities. In contrast, we demonstrated a method to directly measure the wavefunction so that its real and imaginary components appear straight on our measurement apparatus. At the heart of the method is a joint measurement of position and momentum that is made possible by weak measurement (see below for what that is). As an example of the method we experimentally directly measured the transverse spatial wavefunction of a single photon. This new measurement gives the wavefunction a plain and general meaning in terms of a specific set of operations in the lab."
- $\psi$ in configuration space has more degrees of freedom than physical space, therefore cannot correspond to physical reality.
I will never accept the pat line that quantum physics surpasseth all human understanding. We do physics to understand the world, not to shut up and calculate.
There's nothing magical or special about $i$. Complex numbers are just as "real" as Real numbers.
Actually, real numbers aren't real. You can't point to a seventeen. In similar vein complex numbers aren't real either. But the things you count and measure are real. Like apples, and photon wavefunction. Why people ever suggested wavefunction is some kind of probability wave I shall never know. It's surely obvious that $|\psi|^2=\psi\psi^*$ because wavefunction interacts with wavefunction. It takes two to tango.
Both are components of our logical system of computation and together define the number plane - why are physical measurements limited to corresponding to only 50% of the number plane?
I don't think they are. You can measure scalars and vectors. And tensors. Have a look at complex numbers and electromagnetism. If you could give some particular references as to who's saying wavefunction can't be real because complex numbers are involved I'd be interested to read about it.