On the complex plane $\mathbb C$ consider the half-open square $$\square=\{z\in\mathbb C:0\le\Re(z)<1,\;0\le\Im(z)<1\}.$$ Let $p:\mathbb C\to\{0,1,2,3\}$ be any function. Is there a subset $Z\subset\mathbb C$ such that the union of the squares
$$\bigcup_{z\in Z}(z+i^{p(z)}\cdot\square)$$is not Borel in $\mathbb C$.