Let $Q=[0,\infty)\times [0,\infty)\subset \mathbb C$ and $f: Q\times Q\to Q\times Q$ be a diffeomorphism. such that $f$ is holomorphic in the interior of $Q\times Q$. Can we extend this map analytically across the boundary.

Motivation: We have following proposition:

Let $U$ and $V$ are open subsets of $\mathbb R^n_k=[0,\infty)^k\times \mathbb R^{n-k}$ and $f:U\to V$ be diffeomorphism, then

(a). $x\notin \partial U \Leftrightarrow f(x)\notin \partial V$

(b). $f|Int(U)$, and $f|\partial U$ are diffeomorphism.

This proposition gives: If $f:Q\to Q$ is diffeomorphism and holomorphic in the interior. Then either

1- $f$ maps Y-axis to Y-axis and X-axis to X-axis origin goes to origin. OR

2-$f$ maps X-axis to Y-axis and Y-axis to X-axis origin goes to origin.

And using Schwarz reflection principle we have extension in both case. So for $f: Q\to Q$ we have extension across the boundary. I have doubt for $f:Q\times Q\to Q\times Q$.