  We study the sum-product problem for the planar hypercomplex numbers: the
dual numbers and double numbers. These number systems are similar to the
complex numbers, but it turns out that they have a very different combinatorial
behavior. We identify parameters that control the behavior of these problems,
and derive sum-product bounds that depend on these parameters. For the dual
numbers we expose a range where the minimum value of $\max\{|A+A|,|AA|\}$ is
neither close to $|A|$ nor to $|A|^2$.
  To obtain our main sum-product bound, we extend Elekes' sum-product technique
that relies on point-line incidences. Our extension is significantly more
involved than the original proof, and in some sense runs the original technique
a few times in a bootstrapping manner. We also study point-line incidences in
the dual plane and in the double plane, developing analogs of the
Szemeredi-Trotter theorem. As in the case of the sum-product problem, it turns
out that the dual and double variants behave differently than the complex and
real ones.
