Why can't we have a three dimensional system for rotations?

Question

We can have $i^2=-1=j^2$. This allows rotations in the $(1,i)$-plane and $(1,j)$-plane. And I think Euler has a result that rotations about arbitrary axes can be described by successive rotations about the perpendicular $x$, $y$, $z$ axes (correct me if I'm wrong). But this system maybe does not allow rotations in the $(i,j)$-plane because of both $i$ and $j$ squaring to −1 (We haven't defined $i\cdot j$ either). Is this why we don't have a three dimensional number system? If not, then what's the reason we need four dimensions?