Find matrices $X$ such that for any matrix $Y$ we have $\det(X^2 + Y^2) \geq 0$ – math.stackexchange.com

Find matrices $X$ such that for any matrix $Y$ we have $\det(X^2 + Y^2)
\geq 0$ – math.stackexchange.com

What is the characterization of real matrices $X \in \mathbb{C}^{n\times
n}$ such that for any real matrix $Y \in \mathbb{R}^{n\times n}$:
$$\det(X^2 + Y^2) \geq 0?$$