Consider a pair of points, $z_1, w_1 \in \mathbb{D}$, where $\mathbb{D}$ is the unit disc centred at the origin. Is it sufficient to argue that $f(z)=z+a$ (for $a\in \mathbb{C}$) is biholomorphic, since $\exists a$, such that $z_1 + a=w_1$ for any $z_1$ and $w_1$? Or is there a more elegant way to prove this?
Would appreciate your comments.
Your map might not be a map from the unit disc to itself. To prove your original statement, consider the map $w \mapsto \frac{z_0-w}{1-\bar{z_0}w}$ which is a biholomorphism on the disc sending the point $z_0$ to $0$.