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.

Observe that for every $z\in \mathbb C$ the set $(z+i^{p(z)}\cdot\square)$ is the shifted and rotated square $\square$ with a vertex at $z$.

Problem. Is it true that for any function $p:\mathbb C\to\{0,1,2,3\}$ 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 $\mathbb C$?