Question: Which pairs $(n,c)\in\mathbb{Z}_{\ge 1}\times\mathbb{C}$ have the following property:
Let $\sigma$ be the usual cyclic permutation on $1,2,\ldots,n$, and let $z_1,\ldots,z_n,w_1,\ldots w_n\in\mathbb{C}$ such that $$z_{\sigma(i)}=z_i^2+c\quad\text{and}\quad w_{\sigma(i)}=w_i^2+c\quad\text{for all $1\le i\le n$,}$$ as well as $\sum_{i=1}^nz_i = \sum_{i=1}^nw_i$. Then $\{z_1,\ldots,z_n\}=\{w_1,\ldots,w_n\}$.
Examples: The pair $(1,c)$ always has the property.
Why I think this is answerable: There are only finitely many such cycles for a given pair $(n,c)$, since the polynomials $\prod_{i=1}^n(X-z_i)$ must divide $f^{\circ n}-X$, where $f=X^2+c$, and $f^{\circ n}=f(f(f(\cdots)))$, so $f$ iterated $n$ times.