real number line, complex plane, ...?

OK so like we put all the real numbers on a single line, and we put all the multiples of the imaginary unit on a single line, and we put those two lines perpendicular to each other and filled in the area with infinitely many lines that are sums of a real part and an imaginary part, forming the complex plane to represent all complex numbers, but I'd like to have that extended to a 3-space.  So did anyone extend complex numbers with another number line at right angles to the real and imaginary number lines so that we could make a 3-space of infinitely many planes?  Maybe stick in that bothersome "divided by zero" thing, I don't know.

What do you expect from your three dimensional extension of the complex numbers?
Do you want it to be a field? A skew field? Is maybe a vector space enough for you?