Some questions here ask for solutions of equations such as
which then turn out to have no real solutions. However, when complex values are permitted, $\sqrt\cdot$ makes more sense when both leaves are considered and the complex square root
$$\sqrt{z} = \pm\left(\sqrt{|z|+\Re z} + i\,\sigma(\Im z)\sqrt{|z|-\Re z}\right)/\sqrt2$$
is used. Now is there any generalization of the fundamental theorem of algebra how many (if any) complex solutions equations involving radicals like the one above have if (or if not) both leaves are considered?
One related answer mentions the little Picard theorem, which states that a holomorphic function is either constant or takes all but one complex value. However, roots are not entire in $\mathbb C$ and include branch cuts, thus Little Picard doesn't apply here. What we do have here however is a Riemann surface...
