To me, the most fundamental question of quantum physics is this: Why not quaternions? As a mathematician, the distinguishing characteristic of C is that it is the algebraic closure of R; but I see no physical relevance to that.
In general, quantum mechanics is about using complex vectors and unitary operations on them. As it is enough to implement quaternions, they are silently used in quantum mechanics.
The idea is that all the quantum values are complex, and the measurable things are of the form f.f*=|x|^2, you have a symmetry. If you multiply all the universe by i (where i ~= \sqrt(-1) :) ) nothing changes. In general, you can multiply the universe by any complex z such that |z|=1. The global phase is irrelevant. This is the global U(1) symmetry of the universe.
But this symmetry is local. You can choose the phase of each point of the universe independently. The problem is that to compensate the arbitrary picks you have to introduce a new field that compensate for the differences of the local election of the phases. This is just the electromagnetic field. (I'm hiding a technical detail here.) This is the quantum version of the old electromagnetic field of Maxwell. The particles of this field are the photons.
I can't find an "easy" information source (for someone with a math degree), but you may start reading: http://en.wikipedia.org/wiki/Quantum_electrodynamics#Mathema...
The U(1) group has dimension 1, so you have only one "type" of photons. (This "type" is not related to the usual color of the light.) I think it's possible to extend this to the quaternions group of module 1, but it's a 3 dimensional group, so you would have ¿3? photons "types". If all the new "types" of photons are massless, this should be measurable in the experiments. (If the new "types" of photons are heavy, you need to get a bigger particle accelerator to see them, or a theoretical physicist to prove that this idea is actually impossible.)
More technical details: The arbitrary election is called gauge theory, this is a local gauge theory because. The other symmetries of the universe is the SU(3) "color" symmetry that is behind the strong force. It's the color of the quarks inside the nucleus it's not abelian so it's more complicated. The associated particles are the gluons, and there are 8 "types". The other symmetry is the SU(2) that is the cause of the weak force, and the particles are the W+, Z and W-.
Actually the SU(2) symmetry is mixed with the U(1) symmetry. So the real U(1) group create the a version of the electromagnetic field that is related to the weak-hypercharge, and the photon and Z particles are a mix of the particles of the two fields. The mathematical idea is nice, because you have a copy of U(1) inside SU(2), so you can mix both U(1) groups and mix the particles. As an introduction, you can read http://en.wikipedia.org/wiki/Electroweak_interaction#Formula... but if you don't want to dive in the technical details you can think that the U(1) group is (almost) the electromagnetism an live more happily.
> More technical details: The arbitrary election is called gauge theory, this is a local gauge theory because.
What comes after 'because'?
I should ask a specialist (In Physics, I only have a 50% major.) IIRC it's possible to use any Lie group, but they usually like simple connected groups. In particular, they prefer SU(2) to SO(3) in spite they have the same Lie algebra. (It's actually not an arbitrary preference, to describe spin 1/2 particles like the electron, proton, ... you need to use SU(2). The group SO(3) is useful only with integer spin particles.)