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Luke_A_Somers comments on Common sense quantum mechanics - Less Wrong
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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