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Let $a \in \mathbb{C}$, $b \in \mathbb{R}$ and $z \in \mathbb{C}$. Then $z=x+yi$ and $a=c+di$ for some $x,y,c,d \in \mathbb{R}$.\\  So, assume $\left|z^2\right|+Re(az)+b=0$  $\Leftrightarrow x^2+y^2 +Re((c+di)(x+yi))+b =0$ \\  $\Leftrightarrow x^2+y^2 +Re(cx-dy+i(dx+cy)+b=0$