$$\begin{aligned} & z_1+z_2=5 \\ & z_1^3+z_2^3=20+15 i \\ & z_1^3+z_2^3=\left(z_1+z_2\right)^3-3 z_1 z_2\left(z_1+z_2\right) \\ & z_1^3+z_2^3=125-3 z_1 \cdot z_2(5) \\ & \Rightarrow 20+15 i=125-15 z_1 z_2 \\ & \Rightarrow 3 z_1 z_2=25-4-3 i \\ & \Rightarrow 3 z_1 z_2=21-3 i \\ & \Rightarrow z_1 \cdot z_2=7-i \\ & \Rightarrow\left(z_1+z_2\right)^2=25 \\ & \Rightarrow z_1^2+z_2^2=25-2(7-i) \\ & \Rightarrow 11+2 i \\ & \left(z_1^2+z_2^2\right)^2=121-4+44 i \\ & \Rightarrow z_1^4+z_2^4+2(7-i)^2=117+44 i \\ & \Rightarrow z_1^4+z_2^4=117+44 i-2(49-1-14 i) \\ & \Rightarrow\left|z_1^4+z_2^4\right|=75 \end{aligned}$$