[posterboy]

I think it does extend to higher dimension. I have to read up on hypercomplex numbers, Clifford algebra and geometric algebra. Quaternions are known to facilitate arbitrary rotations in 3D. I suppose, complex numbers are more powerful in analysis rather than discrete mathematics, if there is a difference at all.

[ajkjk]

Quaternions model 3d rotations, and Clifford algebras in general model rotations in arbitrary higher dimensions, but it's definitely non-trivial to go from "complex numbers model 2d" to all of that machinery.

What I mean is that the intuition around complex numbers doesn't extend. Certainly not the "i is the square root of -1"-level understanding of complex numbers.

Clifford algebras are better thought of as structures built over the higher-dimension vectors spaces, with little relationship to algebras/fields. They have much more in common with the matrix representation than the algebraic one. The algebraic perspective only supports complex, quaternion, and octonion numbers before breaking down.