Non-trivial squares of bases in Clifford algebras? I've been learning about Clifford algebras recently, and I know that bases can square to +1, -1, or 0 (but are not real numbers themselves).
Is there any value in considering bases that square to other numbers? In particular, are there possibly bases that square to $\pi$, exp(1), and/or the prime numbers (but are not real numbers themselves). Perhaps such bases wouldn't correspond to well-behaved algebras like Clifford algebras. Is there a type of algebra that these bases would fit into, if not Clifford?

Clifford algebras are associated to quadratic forms. Any real non degenerate quadratic form has a basis $(e_1...e_n)$ in which it can be written as $$ q(x_1e_1+...+ x_ne_n) = x_1^2+...+x_r^2 -x_{r+1}^2-...-x_{n}^2$$
for some r between 0 and n.
This is the reason only these quadratic forms are considered in the theory of real Clifford algebras.
