I've been implementing real Clifford algebras and have successfully extended the implementation to include basis vectors that square to $1$, $-1$, and $0$ as well as to include a mix of basis vectors that either commute $(uv-vu=0)$ or anticommute $(uv+vu=0)$ with each other.
I have found that the various permutations are real isomorphic to classic real Clifford, complex Clifford, hyperdual numbers, and many other unital associative algebras ($23$ unique algebras in 3D).
Is this still under the name of Clifford algebras, or is there another name for this extension?