Let $U$ be a Riemann surface and let $z:U\longrightarrow B(0,1)$ be a diffeomorphism, where $B(0,1)$ is the open unit disc in $\mathbf{C}$. So $z$ is a coordinate around $P=z^{-1}(0)$.
Let $Q\in U$ be another point and let $V \subset U$ be an open subset. (We choose $V$ such that its image under $z$ is an open disc around $z(Q)$ contained in $B(0,1)$. A picture would help alot here.) Consider the morphism $z^\prime: V\longrightarrow B(0,1)$ defined by $z^\prime(x) = z(x) - z(Q)$. This is a coordinate around $Q$. Essentially, it is the coordinate at $P$ translated by $z(Q)$.
Question. What is the relation between $dz$ and $dz^\prime$?
My other question The chain rule for a function to $\mathbf{C}$ suggests that they are $\textbf{not}$ the same. (Take $a=z(Q)$ and note that $z^\prime = t_a\circ z$.)