Factored $z^4 + 4$ into $(z^2 - 2i)(z^2 + 2i)$. How to factor complex polynomials such as $z^2 - 2i$ and $z^2 + 2i$? I was wondering how to factor complex polynomials such as  $z^2 - 2i$ and $z^2 + 2i$?
I originally had $z^4 + 4$, which I factored to $(z^2 - 2i)(z^2 + 2i)$ by substituting $X = z^2$ and using the quadratic formula. However, I'm not sure if there exists any method to factor $z^2 - 2i$ and $z^2 + 2i$? 
I would greatly appreciate it if people could please explain how one would go about this.

You can also go brute force, it sometimes has advantages when the polar form is not sympathetic.
$z^2=(x+iy)^2=2i\iff x^2+2ixy-y^2=2i\iff\begin{cases}x^2-y^2=0\\xy=1\end{cases}\iff z=\pm(1+i)$
