jmc

Drawing the square in the complex plane, we see that 4 and $-2+4i$ are opposite vertices. The length of the diagonal is the magnitude of the difference of these numbers, $|4-(-2+4i)|=|6-4i|=\sqrt{6^2+4^2}=\boxed{2\sqrt{13}}$.



Alternatively, we could note that 4 and $3+5i$ are adjacent vertices, so the square has side length $s=|4-(3+5i)|=|1-5i|=\sqrt{1^2+5^2}=\sqrt{26}$. The diagonal then has length $s\sqrt{2}=\sqrt{26}\cdot \sqrt{2}=\boxed{2\sqrt{13}}$.