The two requirements are that it be on the domain of probabilities (reals on 0-1), and that they nest properly.
Quaternions would be OK as far as the Born rule is concerned—why not? They have a magnitude too. If we run into trouble with them, it’s with some other part of QM, not the Born rule (and I’m not entirely confident that we do—I have hazy recollection of a formulation of the Dirac equation using quaternions instead of complex numbers).
The two requirements are that it be on the domain of probabilities (reals on 0-1), and that they nest properly.
Quaternions would be OK as far as the Born rule is concerned—why not? They have a magnitude too. If we run into trouble with them, it’s with some other part of QM, not the Born rule (and I’m not entirely confident that we do—I have hazy recollection of a formulation of the Dirac equation using quaternions instead of complex numbers).
Here are some nice arguments about different what-if/why-not scenarios, not fully rigorous but sometimes quite persuasive: http://www.scottaaronson.com/democritus/lec9.html