complex numbers - Diffusion equation Lagrangian: what is the conjugate field?

Morse and Feshbach state without elaboration that the diffusion equation for temperature or concentration $\psi$ and its "conjugate" $\psi^*$ (quotation marks theirs) has Lagrangian density:

$$L=-\nabla\psi\cdot\nabla\psi^* -\frac{1}{2}a^2(\psi^*\frac{\partial\psi}{\partial t}-\psi\frac{\partial\psi^*}{\partial t}). $$

I don't understand what the conjugate field, $\psi^*$, is. Since the classical (non-Shrödinger) field should be real, I suspect the conjugation symbol * refers to something other than complex conjugation. With a real field, $\psi^*=\psi$, and only $-\nabla\psi\cdot\nabla\psi$ remains, which would be the Lagrangian for the Laplace equation (steady state diffusion).