Find all polynomials $P$ for which $(P(x)-x)\mid P^{(n)}(x)-x$ $n\gt1$ is a fixed natural number.  Find all polynomials $P(x)$ with complex coefficients for which $(P(x)-x)\mid P^{(n)}(x)-x,$ where $P^{(n)}()$ is the $n$th iterate: $P^{(1)}(x)=P(x)$ and $P^{(i+1)}(x)$ = $P(P^{(i)}(x))$
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What I proved until now : I proved $P(x)-x$ does not have any double roots and the problem is equivalent to solving $P(x)-x\mid P^{'}(x)^{n}-1$

Write
$$P[x,y]=\frac{P(x)-P(y)}{x-y}$$
$$P_1(x)=P[x,P(x)]\text{.}$$
Then
$$P^{\circ n}(x)-x=\left(\sum_{k=0}^{n-1}\prod_{i=0}^{k-1}P_1(P^{\circ i}(x))\right)(P(x)-x)$$
witnesses
$$P(x)-x\mid P^{\circ n}(x)-x\text{.}$$
In particular, the ring of scalars is irrelevant.
