Example 3. Let $\approx \in C$. Solve the equation:
$$
\frac{1}{2}(z-1)=\frac{\sqrt{3}}{2}(1+z) i \text {. }
$$
Solve the equation $(1-\sqrt{3} i) z=1+\sqrt{3} i$, knowing $\omega=-\frac{1}{2}+\frac{\sqrt{3}}{2} i$, and $\omega^{2}=-\frac{1}{2}-\frac{\sqrt{3}}{2} i$. Then the equation can be transformed into $\omega z=\omega^{2}$, thus obtaining the solution $z=\omega$.