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.

Solving $$z^2-2i=0$$
we have $$z^2=2i=2\exp\left( \frac{i\pi}{2}\right)$$
$$z=\pm\sqrt2\exp\left( \frac{i\pi}{4}\right)=\pm\sqrt{2}\left(\cos\left( \frac{\pi}{4}\right)+i\sin\left(\frac{\pi}{4} \right)\right)=\pm(1+i)$$
Hence, $$z^2-2i=(z-(1+i))(z+(1+i))$$
Try the same trick on $z^2+2i$.
