Reality of the Wavefunction - Complex numbers and Degrees of Freedom in Configuration space 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:


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*$\psi$ involves complex numbers. Only Real numbers correspond to measurable quantities.

*$\psi$ in configuration space has more degrees of freedom than physical space, therefore cannot correspond to physical reality.
My question is as follows:


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*There's nothing magical or special about $i$. Complex numbers are just as "real" as Real numbers.  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?
This is a follow-on to the question raised in "Reality" of EM waves vs. wavefunction of individual photons - why not treat the wave function as equally "Real"?
 A: 
why are physical measurements limited to corresponding to only 50% of the number plane?

This makes no sense: the real line is a set of measure zero in the complex plane. It does not represent the 50% of it.
If you want to go that way, I would say that $\mathbb R^n$ is "as real" as $\mathbb R$ anyway, so why restrict to $\mathbb C=\mathbb R^2$? why not to use $\mathbb R^3, \mathbb R^4,\cdots$?
The reason to use a complex wave-function is that it allows us to efficiently model the fact that Nature seems to add amplitudes, not probabilities. The physics of this are very well exemplified by the double slit experiment: try to think about that experiment without math. Now try to make up a mathematical model that reproduce the observed characteristics of the experiment. You should convince yourself that complex numbers do the work better than anything else, and this is the reason we want to introduce them.
After all, QM is just a mathematical model that is intended to reproduce observed phenomena. We use the model that works best, and standard QM is the best model we could find. It uses complex number as a mathematical tool, but physics is about modeling Nature, and not about finding the deep internal gears that make Nature work: reality is not mathematical. Nature doesn't work with complex numbers; we humans use them to model Nature.
A: You are correct. The imaginary part of a complex number is no less real than the real part. Both are necessary to give consistent mathematics and because mathematics play such a such an important part in out understanding of nature it is not surprizing that complex numbers appear in our theories of nature.
However, it is an observational fact that all our observations always just give us real values. Can you think any measurement that one can make that gives you a complex number as an answer? When I measure light using a detector I find that the detector always respond in proportion to the intensity of the light. When I do a quantum experiment I count particles. Both the intensity and the particle count are real numbers. 
If I want to use quantum mechanics to predict how many particles I can expect to observe, then I have to compute the propability and that comes out to be the modulus squared $|\psi|^2=\psi\psi^*$ of the complex valued probability amplitude $\psi$. The same applies for the intensity. In this way both the real part and the imaginary part contribute equally. So they are equally important in the calculation.
A: First you may misunderstand what the $\color{red}{\text{quantum state}}$  is.In fact it is a vector in Hilbert space rather than a point in configuration space (About this you can refer the book written by Leonard Susskind(the theoretic minimum:quantum mechanics).So there is not any relation between wave-function and configuration space.And finally you will find the so-called wave-function is nothing but the coefficients before the corresponding base vector.
If you are interested in quantum mechanics,I advise you can follow Leonard Susskind's course.Then you will ask the right question.
Hope this helps.
A: 
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. 
A: The fact that the wavefunctions in QM are represented by complex numbers originates from the basic postulate of QM. That is that the wavefunction alone at t=0 should determine it at other times. This is unlike classical mechanics where both position and velocities at initial time are required in order to predict the evolution of the system. So one can write a wave going in +x direction by exp(ikx). Just multiply by exp(-iwt) and you can get a travelling wave in +x. Similarly if you write exp(-ikx) you know for sure it is a wave that would travel in -ve x direction.
