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\(\displaystyle{ Re( z^{2}) \ge 0}\)

[octahedron]

\(\displaystyle{ z=r(\cos\varphi+i\sin\varphi)\\
z^2=r^2(\cos 2\varphi+i\sin 2\varphi)\\
Re(z^2)=r^2\cos 2\varphi\ge 0\\
\cos 2\varphi\ge 0\\
-\frac{\pi}{2}+2k\pi\le 2\varphi\le \frac{\pi}{2}+2k\pi\\
-\frac{\pi}{4}+k\pi\le \varphi\le \frac{\pi}{4}+k\pi\\
\varphi\in\left(-\frac{\pi}{4};\frac{\pi}{4} \right)\cup\left(\frac{3\pi}{4};\frac{5\pi}{4} \right)}\)

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