While this is an old question, I imagine many encounter geometric algebra through <PERSON> writing on zitterbewegung, so I'd like to offer a different perspective than the existing answers.
Independently of <PERSON>'s zitterbewegung program, his development of Dirac theory provides a clarity to what is being described in the theory that is difficult to accomplish in the usual matrix representation. This is in part due to the geometrical interpretation provided to the imaginary $i$ in the Dirac equation. While I don't know that this interpretation univocally explains the presence of complex numbers in QM, it certainly provides a way to make sense of it.
There are many algebraic objects that square negatively in a geometric algebra, and these generate rotations in distinct planes represented by those objects. So from the geometric algebra point of view, the imaginary $i$ could be thought of as a generator of rotations in a generic plane.
Let's consider a few examples in spacetime, $\mathcal{\text{Cl}}_{1,3}$.
The bivector $\gamma_2 \gamma_1$ squares to $-1$ and generates rotations in the plane spanned by the orthonormal vectors $\gamma_1$ and $\gamma_2$. This acts on vectors in the following way:
$$v \mapsto v' = e^{\gamma_2\gamma_1\phi/2} v e^{-\gamma_2\gamma_1\phi/2}.$$
Or consider the pseudoscalar $I$ which generates rotations between multivectors and their Hodge duals. 364
In electrodynamics, the pseudoscalar generates duality rotations of the <PERSON> bivector $F$ via: $$F \mapsto F' = e^{I\beta/2} F e^{I \beta/2}.$$
(Yes without the minus — the reason is the general transformation law $M' = \psi M \tilde \psi$ involves reversion $\tilde \psi$ that behaves differently for these two examples: $\widetilde{\gamma_2\gamma_1} = - \gamma_2\gamma_1$ but $\tilde I = I$.)
As an aside (in response to <PERSON>'s comment), the symmetric stress-energy tensor in electrodynamics is given by
$$T(n) = \frac{1}{2} F n \tilde F,$$
which represents the flow of energy-momentum through the surface perpendicular to $n$, and describes a reflection of $n$ across the bivector $F$. I personally find this to be clearer and more manageable than:
$$T^{\mu\nu} = F^{\mu\alpha}F^{\nu}{}{\alpha} - \frac{1}{4} \eta^{\mu\nu}F{\alpha\beta}F^{\alpha\beta} = \gamma^\mu \cdot T(\gamma^\nu),$$
so I think it's misleading to say that this formalism "has a hard time with symmetric tensors," even if tensors form a larger algebra.
There are also bivectors that square positively and generate hyperbolic rotations (Lorentz boosts) in spacetime. For instance, take the timelike bivector $\gamma_3 \gamma_0$, which squares to $+1$. This boosts vectors in the $\gamma_3$ direction in the "$\gamma_0$-frame" as:
$$v \mapsto v' = e^{\gamma_3\gamma_0\phi/2} v e^{-\gamma_3\gamma_0\phi/2}.$$
General <PERSON> transformations are of the form $R = e^{B/2}$ where $B$ is a spacetime bivector, with vectors transforming as
$$v \mapsto v' = R v \tilde R = e^{B/2} v e^{-B/2}.$$
Here's the point. The dynamical quantity of Dirac theory (a spinor) is precisely a dilation, duality, and Lorentz transformation given by
$$\psi = (\rho e^{I \beta})^{1/2} R. 364