= Solution
In polar coordinates, $z=re^{i\phi}$ and
$$
L=-i\hbar\frac{\partial}{\partial\phi}.
$$
Consequently
$$
L\psi=\hbar\psi,
\qquad
L\psi^*=-\hbar\psi^*.
$$
The two states are orthogonal. They are degenerate energy eigenstates, so their coefficients acquire the same overall time-dependent phase; equivalently, $[L,H]=0$ makes $\langle L\rangle$ constant. For the normalized state
$$
\frac{\sqrt5}{3}\psi+\frac23\psi^*,
$$
the <angular momentum of a complex-coordinate Gaussian state> is therefore
$$
\boxed{
\langle L\rangle_t
=\frac59\hbar-\frac49\hbar
=\frac\hbar9},
\qquad t>0.
$$
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