> Similarly, to rotate vectors you have to create matrices, which don't exist in ℝ2, and apply them to vectors through matrix multiplication.

You don't "need" matrices, they're just one way to represent linear functions that map vectors, and the complex field is another (for R^2).

Coincidentally, most of the concrete implementation described for operations in G^2 here are applications of Euler's formula.

This article drops off in extending the lessons from R^2 to R^3. Comparatively, matrices excel for comprehensibility in this regard -- if you know how to apply matrices in R^N, you can apply the same knowledge to R^N+1.