Does Quantum Mechanics need imaginary numbers? [duplicate]

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In quantum mechanics, we assume wavefunctions are complex valued, and that probability amplitudes are given by the modulus of the wavefunction squared. This formalism can correctly explain interference effects, e.g. the double-slit experiment. I understand all of this.

Now, let's consider classical optics. If we have a linearly polarized EM wave, the electric field is defined in all space by a single real number. If we want to find intensity of light at any given point, we time-average the square of the electric field at that point. Alternatively, instead of representing the electric field with a time-varying real number at every point, we can also represent the electric field by a complex phasor which rotates in time. Then then intensity of the light at any point is (up to a factor of $\sqrt{2}$) just the modulus of the phasor squared. And if the light interferes with itself (e.g. in an interferometer) the new phasor at the point of interference is simply the sum of the old phasors, identical to interference in quantum mechanics.

So my question is: if, instead of assuming we have a complex valued wavefunction, we assume we have a REAL wavefunction (which is just the real part of our normal wavefunction), and all our probability measurements are inherently time-averaged (like our intensity measurements for an EM wave), would we run into any contradictions with logic/experiment?

(I realize this wouldn't be a particularly USEFUL formulation of quantum mechanics. Complex numbers are obviously more elegant--we invented phasors for a reason, after all. I just want to know if it's equivalent.)

Edit: I realize complex numbers can be thought of as simply two real numbers. I'm interested in the case of representing the wavefunction as a SINGLE real number. I guess what I'm looking for is a concrete difference between wavefunctions and electric fields.