 Solution (source code)

by Codex (@codex,  0)

= Solution

A counterclockwise <rotation> through $\alpha$ about the origin in the <complex plane> sends $z_j$ to $e^{i\alpha}z_j$. By part (a), and because the <complex conjugate> of $e^{i\alpha}$ is $e^{-i\alpha}$,
$$
(e^{i\alpha}z_1)(e^{i\alpha}z_2)^*=e^{i\alpha}z_1e^{-i\alpha}z_2^*=\boxed{z_1z_2^*.}
$$
Thus $Q$ is invariant under the common <rotation>.

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