Florin,
It is nice to hear from you. It has unfortunately been a while since I went to your website. I had to reinstall browser and lost the bookmark.
The example with quaternionic QM is interesting. This is the quaternion Dirac equation and the model of the H-atom is a Euclidean form of gravitation. The complex wave function for a system is equivalent up to a phase, and for quaternions they are equivalent up to multiplication by any quaternion, not just the unit quaternion --- at least as I recall. We could write the quaternion wave function as
Ψ = c_1i + c_2j + c_3k + c_41
Expanded in the quaternion basis i, j, k, 1. This is equivalent under multiplication by a quaternion, just as the wave function gives equivalent physics when multiplied by a c-number or phase. So for simplicity multiply this by the quaternion i to get
iΨ = c_1i*i + c_2i*j + c_3i*k + c_4i
= -c_11 + c_2k - c_3j + c_4i
which is just an SO(4) rotation in the basis of elements. Of course QM and physics is invariant under changes of coordinates.
In point of fact the Dirac operator and the quaternion function are both the same thing --- quaternions. The action of the Dirac operator on the quantum field quaternion is equivalent to the cohomology condition ψψ = 0. This is known as the Pauli exclusion principle.
The statement that the wave function is real does have this odd implication that all complex numbers are equal to a quaternion. Mathematically this is strange, and even without quaternions the wave function being real means complex numbers are all real. Physically the nonlocal properties of QM simply can't be reduced to a classical realization. We might even go so far as to say that classical physics is an illusion. We know it is an illusion because it is falsified outside of certain domains of applicability, such as atomic physics.
Cheers LC