$\begin{aligned} & \text x^4+x^2+1=0 \\ & \left(x^2+x+1\right)\left(x^2-x+1\right)=0 \\ & x^2+x+1=0 \text { have roots } \omega, \omega^2 \\ & x^2-x+1=0 \text { have roots }-\omega,-\omega^2 \\ & \omega+\omega^2=-1 \Rightarrow \alpha=\omega, \beta=\omega^2 \\ & -\omega-\omega^2=1 \Rightarrow \gamma=-\omega, \delta=-\omega^2 \\ & \therefore \quad \alpha^{2023}+\beta^{2023}+\gamma^{2022}+\delta^{2022} \\ & \quad=(\omega)^{2023}+\left(\omega^2\right)^{2023}+(-\omega)^{2022}+\left(-\omega^2\right)^{2022} \\ & \quad=\omega^{2023}+\omega^{4046}+\omega^{2022}+\omega^{4044} \\ & \quad=\omega+\omega^2+1+1=1 .\end{aligned}$