According to this question (Ritt's classification), given two commuting polynomials $f,g \in \mathbb{C}[z]$, there exists a common fixed point of them. By commuting we mean: $f(g(z))=g(f(z))$.
Of course, a polynomial $h(z)$ over $\mathbb{C}$, not of the form $z+\lambda$, $\lambda \in \mathbb{C}^*$ has a fixed point, since there is a solution in $\mathbb{C}$ to $h(z)=z$; notice that for this claim $\mathbb{C}$ can be replaced by any algebraically closed field; see this question.
Is it possible to replace $\mathbb{C}$ by any algebraically closed field of characteristic zero in the common fixed point result?
Any hints are welcome!
Yes, it holds for any algebraically closed field of characteristic $0$. Since the statement can be formulated in first-order logic, all you have to do is to apply the Lefschetz principle.