As I understand it, complex numbers: $ax+i$ are isomorphic to two-dimensional space.
Quaternions consist of $4$ dimensions. Is that right? Wikipedia says "quaternions form a four-dimensional associative normed division algebra over the real numbers."
Is there a mathematical system generalizing complex numbers that consists of $3$ dimensions and is isomorphic to 3-dimensional space?
Not completely an answer to your question, but this might be interesting to you:
The Frobenius theorem states that up to isomorphism there are three finite-dimensional (unital) associative division algebras over the reals: the reals themselves (dimension 1), the field of complex numbers (dimension 2), and the quaternions (dimension 4).
If you're willing to give up associativity you can also add the octonions (dimension 8) to that list. See Hurwitz's theorem for that.
So depending on how similar to the complex numbers you want it to be, the answer might be a definitive no.