I met a problem, which says if $f$ and $g$ are all entire functions on $C$. And $f(g(z)) \in C/[- \infty，0]$, for all $z \in C$. Then both $f$ and $g$ must be a constant.
I have no idea about how to reach this conclusion, is it about the CR equation or I need to expand them into some series?
The correct conclusion is that at least one of $f$ and $g$ is constant. You cannot conclude that both are constant. Write $\psi= f \circ g$. Let $h$ be a conformal injective map from ${\Bbb C} \setminus (-\infty,0]$ to the unit disk. This exists by the Riemann mapping theorem; for an explicit example, fix a branch of the square root function that is well defined on ${\Bbb C} \setminus (-\infty,0]$ and take $$h(z)=\frac{\sqrt{z}-1}{\sqrt{z}+1}\, .$$ Then $h \circ \psi$ is bounded; by Liouville [1] it is constant, whence $\psi$ is also constant. If $g$ is not constant, its image must contain a continuous curve on which $f$ is constant, so $f$ equals a constant everywhere.
[1] https://en.wikipedia.org/wiki/Liouville%27s_theorem_(complex_analysis)