It really doesn't help that in 2D you can't do much that you can't do with complex numbers. I'm not a mathematician and I really struggle with just accepting t

It really doesn't help that in 2D you can't do much that you can't do with complex numbers. I'm not a mathematician and I really struggle with just accepting the mathematical definition. I've found it helpful to think of the geometric algebra as an extension of the R^2 to contain scalars and geometrically well-defined imaginary numbers. In R^2, the dot product maps to R. You can't store this in R^2. You can store this in the corresponding geometric algebra. For R^3, the corresponding geometric algebra forms an 8-dimensional space: 1 for the scalars, 3 for the vectors, 3 for the bi-vectors (planes formed by the 3 basis vectors) and 1 as the geometrically meaningful imaginary part. And yet in 3D and beyond, geometric algebra provides a really elegant framework for rotations in a way that is consistent, self-contained, and readily extends to higher dimensions.