jmc

First of all, we know that $|ab|=|a|\cdot |b|$, so $$\left|{\left(1+\sqrt{3}i\right)}^4\right|={\left|1+\sqrt{3}i\right|}^4$$We also find that $$\left|1+\sqrt{3}i\right|=\sqrt{{\left(1\right)}^2+{\left(\sqrt{3}\right)}^2}=\sqrt{4}=2$$Therefore, our answer is $2^4=\boxed{16}$.