Given equation is $x^2-2 x+2=0$
$$
\begin{aligned}
& \therefore \alpha+\beta=2 \text { and } \alpha \cdot \beta=2 \\
& \because(\alpha-\beta)^2=(\alpha+\beta)^2-4 \alpha \beta=4-8=-4 \\
& \alpha-\beta= \pm 2 i \\
& \alpha+\beta=2
\end{aligned}
$$
On solving equations (i) and (ii)
We get $\alpha=1+i$ and $\beta=1-i$
$$
\begin{aligned}
& \therefore \alpha^2=(1+\mathrm{i})^2=2 \mathrm{i} \Rightarrow\left(\alpha^2\right)^2=-4 \\
& \beta^2=(1-\mathrm{i})=-2 \mathrm{i} \Rightarrow\left(\beta^2\right)^2=-4 \\
& \text { Now, } \alpha^{2020}+\beta^{2020}=\left(\alpha^4\right)^{505}+\left(\beta^4\right)^{505} \\
& =(-4)^{505}+(-4)^{505}=-\left[2^{1010}+2^{1010}\right] \\
& =-2.2^{1010}=-2^{1011}
\end{aligned}
$$