 Solution

Rearrange to $z^4=1+i\sqrt{3}=2e^{i\pi /3}$. A fourth root has modulus $2^{1/4}$ and an argument one quarter of $\pi /3+2\pi k$. Therefore all four solutions areTheir arguments differ by $\pi /2$, so they are distinct. Raising any of them to the fourth power returns $1+i\sqrt{3}$.