The answer is "no":
Take $M>2$ and evaluate
$$\mathbb{E}[|z_1|^2]=\mathbb{E}[|z_2|^2]=\frac{1}{2M(M-1)},$$ $$\mathbb{E}[|z_1|^2|z_2|^2]=\frac{1}{4M(M+1)(M-1)(M-2)},$$ $$\Rightarrow\mathbb{E}[|z_1|^2|z_2|^2]-\mathbb{E}[|z_1|^2]\mathbb{E}[|z_2|^2]=\frac{1}{2M^6}+{\cal O}(M^{-7}),$$ so $z_1$ and $z_2$ are correlated no matter how large $M$.