In this talk, I will discuss the unit distance and distinct distances problems over the planar hypercomplex numbers: the dual numbers D and the double numbers S. I will show that the distinct distances problem in S^2 behaves similarly to the original problem in R^2. The other three problems behave rather differently from their real analogs. We can study those three problems by introducing various notions of the multiplicity of a point set. The analysis is based on an understanding of the geometry of the dual plane and of the double plane. We will also rely on classical results from discrete geometry, such as the Szemerédi-Trotter theorem.”