I’ve written a number of posts about **four-dimensional Euclidean space**, usually in the context of one of my favorite geometrical objects, the **tesseract**. I’ve also mentioned 4D Euclidean spaces as just one of many possible multi-dimensional parameter spaces. In both cases, the familiar 2D and 3D spaces generalize to additional dimensions.

This post explores a specialized 4D space that uses complex numbers along each axis of a 2D nominally Euclidean space. Each **X** & **Y** coordinate has two degrees of freedom, a *magnitude* and a *phase*. This doesn’t make 4D spaces easier to visualize, but it can offer a useful way to think about them.

It also connects back to something I wrote about in my QM-101 series.