First, I suggest a small but very important correction to your statement, "imaginary numbers have no natural ordering." You must mean, "*complex* numbers have no natural ordering." Purely imaginary numbers alone cannot be distinguished from the purely real numbers without a rotation operator or some relation enforcing independence; thus, taken alone, they are ordered. A physical problem gives meaning to the symmetry. Refer to the dispersion and attenuation of x-rays for an example.
Second, I sympathize with your concern regarding the treatment of the complex numbers in quantum mechanics. However, your complaint gives a superficial appearance by your reference to "complicated." It sounds as though you are complaining about difficulty rather than validity. On the contrary, I did read your PDF note, and I believe you are "on to something," perhaps the same thing at which I am digging... read on.
Third, we must carefully voice our concerns without any superstition or fear about the complex numbers. We know "complex" does not mean "complicated" as in "difficult," but rather, the term implies "augmented." The union of two similar but independent parts forms a "complex," a complex number. OK, with that caveat stated, we can proceed.
Finally, I'll reiterate the complex-number thought-problem with Schrodinger's equation: we axiomatically assume a monotonic (entropic) time evolution in quantum mechanics. Schrodinger's equation is often written with the time-forward energy on the left and the Hamiltonian operator on the right (of the equality). This mathematical statement requires the time-evolution of energy from "real" to "imaginary." The rotation thus implied neglects the reciprocity of nature. This is a diffusion/heat equation rather than a wave equation. Though time appears irreversible to us (as it must, according to the causal biophysics of memory - the anthropic principle), how can we forbid its reversal? Can we really believe nature is non-reciprocal at a fundamental level? If energy is conserved, and yet we allow it to flow, we ought admit the ebb solution.
The world once was believed flat, but science and exploration found its great circle. If time is presently believed monotonic, when will we look for a great circle?