Dual numbers and double numbers are also planar.
Don't know how relevant this is, but I've been thinking about a better naming scheme for hypercomplex number systems. I came to it after seeing a paper about the "dual-complex numbers", which are not a straightforward complexification of the dual numbers as one might expect. Hopefully, the scheme should be pronounceable, and without the possibility of confusing it for something else. This town needs law! I'm thinking of asking for suggestions for what it should be exactly.