Elementary Incidence Theorems for Complex Numbers and Quaternions

We present some elementary ideas to prove the following Sylvester-Gallai type theorems involving incidences between points and lines in the planes over the complex numbers and quaternions. 1. Let $A$ and $B$ be finite sets of at least two complex numbers each. Then there exists a line $\ell$ in the complex affine plane such that $\lvert(A\times B)\cap\ell\rvert=2$. 2. Let $S$ be a finite noncollinear set of points in the complex affine plane. Then there exists a line $\ell$ such that $2\leq \lvert S\cap\ell\rvert \leq 5$. 3. Let $A$ and $B$ be finite sets of at least two quaternions each. Then there exists a line $\ell$ in the quaternionic affine plane such that $2\leq \lvert(A\times B)\cap\ell\rvert \leq 5$. 4. Let $S$ be a finite noncollinear set of points in the quaternionic affine plane. Then there exists a line $\ell$ such that $2\leq \lvert S\cap\ell\rvert \leq 24$.