Let $f : \mathbb{C} \longrightarrow \mathbb{C}$ be an entire function such that $f(z+1) = f(z+\imath) = f(z)$ for every $z \in \mathbb{C}$. Choose the correct statement(s) from below:\\
(A) $f$ is constant;\\
(B) $f(z) = 0$ for every $z \in \mathbb{C}$;\\
(C) There exist complex numbers $a, b$ such that for every $x, y \in \mathbb{R}$, $f(x + \imath y) = a\sin(x) + \imath b\cos(y)$;\\
(D) $f$ is not necessarily constant but $|f(z)|$ is constant.