Find all roots of $z^4 + 1 = 0$

Question

$z^4 + 1 =0$, looking at a relatively similar question I concluded that the next step is to factorise into $(z^2 + i) (z^2 - i) = 0$. However, I'm not sure what the next step should be? Do i continue to factorise?

Answer 1

$$z^4=-1+0i$$ $$\to z^4=\cos[(2k+1)\pi]+i\sin[(2k+1)\pi]$$ $$\to z= \cos\bigg[\frac{(2k+1)\pi}{4}\bigg]+i\sin\bigg[\frac{(2k+1)\pi}{4}\bigg]$$

$$k \in \Bbb Z$$

And go from there, by inserting $k=0,1,-1$ etc