Consistency depends on your set of axioms. If you are willing to give up various nice properties of division, then you can obviously extend it however you like.

Consistency depends on your set of axioms. If you are willing to give up various nice properties of division, then you can obviously extend it however you like. My gripe with arbitrary choices like this is that it pushes complexity from your proof's initial conditions ("we assume x != 0") into the body of your proof (every time you use division now you've split your proof into two cases). The former is a linear addition of complexity to a proof, whereas the latter can grow exponentially. Of course, nothing is stopping you from using an initial condition anyway to avoid the case splitting, but if you were going to do that why mess with division in the first place?