Show that $f$ has exactly one zero on the square $Q =$ {$x + iy ∈ \Bbb C : |x| < 1, |y| < 1$}. Let $f(z) = z + g(z)$ where $g$ is holomorphic. Suppose that
$|\operatorname{Im} g(z)| < 1$ for $z ∈ [−1 − i, 1 − i]∪[−1 + i, 1 + i]$ and $|\operatorname{Re} g(z)| < 1$ for $z ∈ [−1 − i, −1 + i] ∪ [1 − i, 1 + i]$. 
Show that $f$ has exactly one
zero on the square $Q =$ {$x + iy ∈ \Bbb C : |x| < 1, |y| < 1$}.
My attempt:
I let $h(z) = z$. Then, I want to compare $|g(z)|$ and $|h(z)|$ because if $|g(z)| < |h(z)|$ then by Rouché's theorem, $h$ and $h+g$ have the same number of zeros, and $h$ has in fact one zero. But then $h+g = f$ and thus $f$ would also have the same number of zeros as $h+g$ which has one zero.
This is what I could come up with:
$|g(z)| = |u(z) + iv(z)|$. Then for $z \in Q$, we have $|g(z)| \leq |u(z)| + |v(z)| < 1 + 1 = 2$ (since $|\operatorname{Im} g(z)| < 1$ for $z ∈ [−1 − i, 1 − i]∪[−1 + i, 1 + i]$ and $|\operatorname{Re} g(z)| < 1$ for $z ∈ [−1 − i, −1 + i] ∪ [1 − i, 1 + i]$)
But I don't know how to continue from here. Any help please? 

For this problem the following stronger version of Rouche works (sometimes it is called the symmetric Rouche and is expressed as $|f-g| <|f|+|g|, z \in K$): 
If $\Omega$ is the interior domain of a Jordan curve $K$ and $f(z)+\lambda h(z) \ne 0, \lambda \ge 0, h(z) \ne 0, z \in K$ then $f,h$ have the same number of zeroes inside $\Omega$.
The hypothesis of the OP shows that for $\lambda \ge 0, \Re (f+\lambda z) \ne 0$ when $\Re z = \pm 1$ and $\Im (f+\lambda z) \ne 0, \Im z = \pm 1$ so $f+\lambda z \ne 0$ on the boundary of the square for any $\lambda \ge 0$ while $z \ne 0$ there clearly, so $f,z$ have same number of zeroes inside the square as the OP predicted.
The stronger version of Rouche follows because the homotopy $tf(z)+(1-t)h(z), 0 \le t \le 1, z \in K$ avoids zero by hypothesis ($t=0$ is $h \ne 0$, $1 \ge t>0$ is $f+\frac{1-t}{t}h \ne 0$) so the winding number of $tf(z)+(1-t)h(z)$ around $K$ exists and is continuos for $0 \le t \le 1$ but it is then constant being an integer; at the two ends we get the number of zeroes inside $K$ of $f$ and $g$ respectively
