I need to find a general formulae for a square, with its interior included, in terms of complex numbers. Note that your general square should have (general centre, side-length and orientation.)
I do not know how to deal with the orientation i.e. when the square is 45 degrees.
My solution is that let $a,b,c\in\mathbb{R}$ and $a< b$, then the general square and its interior is the intersection of $a\leq\text{Re}(z)\leq b$ and $c\leq \text{Im}(z)\leq c+|b-a|$ but this only has one orientation.
A square with center the origin and sides, of length $2a>0$, parallel to the axes: $$\max\{|Re(z)|,|Im(z)|\}\leq a.$$
Rotating the square around the origin with angle $\theta$: $$\max\{|Re(z/e^{i\theta})|,|Im(z/e^{i\theta})|\}\leq a.$$
Translating it: $$\max\{|Re((z-c)/e^{i\theta})|,|Im((z-c)/e^{i\theta})|\}\leq a.$$
Using that $Re(z)=(z+\overline{z})/2$ and $Im(z)=(z-\overline{z})/2i$
$$\max\{|(z-c)/e^{i\theta}+\overline{(z-c)/e^{i\theta}}|,|(z-c)/e^{i\theta}-\overline{(z-c)/e^{i\theta}}|\}\leq 2a$$