Claims About a Complex Root Locus

Let $z,w\in\mathbb C$, and suppose the equation $w^2+zw+z\mathrm i=0$ (in $w$) always has a real root. Let $\Gamma$ be the locus of the point $Z$ in the complex plane $xOy$ corresponding to $z$. Determine, with proof, which of the following are true of $\Gamma$.

1. It is symmetric about the origin. 2. It lies below the line $y=1$. 3. It is symmetric about the $y$-axis. 4. It is a closed curve.