Could we define it as a one to many function that maps from 0/0 to the set of all Complex numbers?

If 0/0 = Q, then

0 = 0 × Q. So, we get to define this operation. If we treat 0 as an integer, then we could treat Q (a set) like a 1 x inf. matrix and we have scalar multiplication.

But

0 × Q = [0]' × Q

So 0 can be an infinite set of 0. This is cyclical, I know. This cyclical nature leads to an interesting effect if we also define integer division. Try this around 0 in Q.

We could also define the × as an inner (dot) product, or as a cross product.

This is fun!

Is there a number set larger than a complex number? If so you can’t map to any concrete thing as it is really anything.

And I guess it is easy to define one more dimension of number.