Aaronson actually answers this question directly towards the end of the lecture. The short answer is that quaternions don't work for the same reason that real numbers don't work: the probabilities don't work out right for unitary transformations.

You can represent two-state systems (qubits) with quaternions.

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.

likely because the probabilities in QM need to be commutative (QM operators are not commutative, but the probabilities themselves are).

I remember reading things online where people claim that the natural world (probably including what's known in quantum mechanics) is better explained with quaternions.