Let $S_k=\left\{z \in \mathbb{C}: z^2+|z|=k\right\}$ for each $k \in \mathbb{Z}$. Then

• Correct answer
  A

  $S_0$ has more elements than any $S_k, k \neq 0$
• Correct answer
  B

  for $k \gt 0$, if $z \in S_k$ then $\operatorname{Im}(z)=0$
• Correct answer
  C

  $\left|\operatorname{Arg}\left(z_1\right)-\operatorname{Arg}\left(z_2\right)\right|=\pi$ for $z_1, z_2 \in S_k$ for any $k \neq 0$
• D

  for $k \gt 0$, if $z \in S_k$ then $\operatorname{Re}(z)=0$

Hence, the correct answer is options: 'A', 'B', 'C'.