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A canonical basis for the geometric algebra $\mathcal{G}(3,0)$ has $2^3$ elements:

$1, e_1, e_2, e_3, e_1e_2, e_1e_3, e_2e_3, e_1e_2e_3$

That's easy to understand. There are 3 dimensions and therefore $2^3$ combinations of those 3 dimensions.

But then I read that:

Clifford algebras $Cℓ_{p,q}(C)$, with $p + q = 2n$ even, are matrix algebras which have a complex representation of dimension $2^n$.

(from https://en.wikipedia.org/wiki/Clifford_algebra#Spinors)

Obviously the canonical basis of $Cℓ_{p,q}(C)$ should have $2^{2n}$ elements. Am I to understand that this "complex representation" is somehow a matrix with $2^n$ rows and $2^n$ columns and therefore $2^{2n}$ entries?

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No I guess it is similar to the complex representation of quaternions

$ \begin{bmatrix} a+bi & c+di \\ -(c-di) & a-bi \end{bmatrix}$

I guess that would be considered to be 2 dimensional and therefore $2^1$ elements even though quaternions work in 3 dimensions of space

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