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 $2\leq \lvert S\cap\ell\rvert \leq 24$.