Let us consider a complex function $f$ defined in a set $D$. The function $f$ has a fixed point if the equation $f(s)=s$ have a solution. In the real case, the existence of a fixed point means geometrically that two curves intersect: the curve of $f$ and the line $y=x$.
My question is:
(1) What is the geometrical meaning of the existence of a fixed point for the complex case.
(1) What is the geometrical meaning of the fact that a given point $z$ is not a fixed point for a given complex function $f$.
For a function $f:\Bbb C\to\Bbb C$, the geometric meaning of a fixed point is somewhere where the graph of $f$ (which is a surface) intersects the plane $x=y$ in $\Bbb C^2$. This is not easy to visualize, as our intuitive understanding of space is inherently one dimension too small, but math doesn't care about our ability to imagine.