How can I find ALL complex roots of $z^4 + 1 = 0$? For $z \in \mathbb{C}$, how can I find all roots of the equation
$$
z^4 + 1 = 0
$$
Obviously, this equation implies
$$
z^4 = -1 = e^{i \pi}
$$
and thus, one of our roots must be
$$
z_1 = e^{\frac{i \pi}{4}}
$$

However, I am aware that this is not the only root. How can I find all other roots of this equation?
Actualy you should write

$$z^4=-1=e^{i(\pi+2k\pi)}\to z_k=e^{\frac{i\pi(2k+1)}{4}}$$ 

Now make $k=0,1,2,3$ and get your roots.
If $z=\mathrm e^{i\theta}$, you just have to solve
$$4\theta\equiv \pi\mod2\pi\iff\theta\equiv \frac\pi4\mod\frac\pi 2.$$

Alternatively, once you have one $4$th root of a complex number, you obtain all the roots multiplying it by the $4$th roots of unity, i.e. by $1,-1, i,-i$.