[nyssos]

"Geometric algebras" are Clifford algebras over the reals. Differential geometry is probably the field where you're most likely to see mathematicians discuss them, though they also come up in certain (closely related) areas of mathematical physics via spinors.

[koolala]

What about Clifford algebras over the imaginaries? Is that included in your idea of reals?

[nyssos]

Imaginary numbers aren't a field, so there's no such thing. Clifford algebras over the complex numbers work fine, but it's usually not what the people talking about "geometric algebra" are doing.