  Expressing the Schroedinger Lagrangian ${\cal L}$ in terms of the quantum
wavefunction $\psi=\exp(S+{\rm i}I)$ yields the conserved Noether current ${\bf
J}=\exp(2S)\nabla I$. When $\psi$ is a stationary state, the divergence of
${\bf J}$ vanishes. One can exchange $S$ with $I$ to obtain a new Lagrangian
$\tilde{\cal L}$ and a new Noether current $\tilde{\bf J}=\exp(2I)\nabla S$,
conserved under the equations of motion of $\tilde{\cal L}$. However this new
current $\tilde{\bf J}$ is generally not conserved under the equations of
motion of the original Lagrangian ${\cal L}$. We analyse the role played by
$\tilde{\bf J}$ in the case when classical configuration space is a complex
manifold, and relate its nonvanishing divergence to the inexistence of
complex-analytic wavefunctions in the quantum theory described by ${\cal L}$.
