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 $P^{(i+1)}(x)$ = $P(P^{(i)}(x))$
.
What I proved until now : I proved $P(x)-x$ does not have any double roots and the problem is equivalent to solving $P(x)-x\mid P^{'}(x)^{n}-1$

If $r$ is a root of $P(x)-x$ of order $m$, i.e. $P(x) = x + O((x-r)^m)$ as $x \to r$, then
I claim $P^{(n)}(x) = x + O((x-r)^m)$ as well.  This should be possible to prove by induction on $n$.  Therefore all roots of $P(x) - x$ are roots of $P^{(n)}(x) - x$ with the same or greater multiplicity.  We conclude that $P(x) - x$ always divides $P^{(n)}(x) - x$.  
