Challenge:Complex reducible to quadratic equation: Solve $$z^4+2\left(a^2-b^2\right)z^2+\left(a^2+b^2\right)^2=0$$ where $a,b\in\mathbb{R}$

Solve $$z^4+2\left(a^2-b^2\right)z^2+\left(a^2+b^2\right)^2=0$$ where $a,b$ are real numbers.

Hint 1 : Put $w=z^2$ and find $w$ first.  Then for each $w$, solve $z^2=w$.

Hint 2: see other posts on this label.

Answer: $z=\pm (a+ib), \pm (a-ib)$