Typically GA people are working with real algebras, meaning the coefficients are real, and things like a square root of -1 appear as some object in the algebra (like a 2-blade). But you could also have a Clifford algebra with coefficients in e.g. the complex numbers or fields of finite characteristic.
In fact using different coefficient rings is one way to write a compact recursive definition of real Clifford algebras:
As this shows, ei and ej are imaginary numbers. This confusion is the biggest issue in GA to me. Typically Linear Algebra users don't use imaginary numbers either. It's all connected when doing geometry though and its pointless to draw invisible lines between these concepts.
We have no way to talk about more general types of complex / imaginary numbers besides rigid math lingo that provides no geometric intuition or grace for geometric imagination.
In fact using different coefficient rings is one way to write a compact recursive definition of real Clifford algebras:
http://blog.sigfpe.com/2006/08/geometric-algebra-for-free_30...