the crossover point of four complex points

Question

If there is four complex points $z_1,z_2,z_3,z_4$ in complex plane $\mathbb{C}$, I want to get the crossover point of the line $z_1z_2$ and $z_3z_4$. If I use the $Re(z_i)$ and $Im(z_i)$, it is easy to get the formula of the crossover point, but I want to know if there is a formula of the crossover point without $Re(z_i)$ and $Im(z_i)$? Thanks in advance.

Answer 1

If you mean a function $f:\Bbb C^4\to\Bbb C$, such that the restrictions to one variable are holomorphic, it doesn't exist.

Note that if you move for example $z_1$ along the line $z_1z_2$, the image of $f$ shouldn't change, that is, the function $f_1(w)=f(w,z_2,z_3,z_4)$ should be costant over the line $z_1z_2$. But if $f_1$ is holomorphic, this implies that $f_1$ is constant in the whole plane, and this is not what you want.