Why are complex numbers needed to define qubits?
You can run any standard quantum algorithm on a real-amplitude quantum computer with one additional qubit and only constant-factor slowdown (or perhaps linear-factor slowdown considering loss of parallelism) by replacing each $a{+}bi$ in your unitary matrices by $\big(\begin{smallmatrix}a&amp;-b\\b&amp;a\end{smallmatrix}\big)$. Likewise, you can simulate a quaternionic quantum computer on a standard quantum computer with one additional qubit by replacing $a{+}bi{+}cj{+}dk$ with $\big(\begin{smallmatrix}a+bi&amp;-c+di\\c+di&amp;a-bi\end{smallmatrix}\big)$. So these models are all polynomially equivalent.
I doubt that there's a good non-physical argument for the appearance of complex numbers. In physics they're related to the duality between zeroth and first derivatives of position in harmonic oscillation (the same reason that complex numbers are used in modeling AC circuits), and that duality doesn't exist in the discrete quantum circuit model. Another argument from physics is that in Lagrangian QM the amplitude of a transition is $\int e^{iS(γ)} \, \mathrm dγ$ (where $γ$ ranges over &quot;histories&quot;), and there doesn't seem to be anything you can sensibly replace $i$ with in the real case. But this model, where each history contributes an amplitude with the same absolute value but a different phase, only works because of cancellation of nearby histories, and there aren't any &quot;nearby&quot; histories in the quantum circuit model.