Consider a complex scalar field $\psi(x)$ with Lagrangian density
$$ \mathcal{L} = \partial_\mu\psi^* \partial^\mu\psi - M^2\psi^*\psi. $$
From the Lagranigan density we obtain the momentum $\pi = \dot\psi^*$. How do you obtain this? I expanded $\partial_\mu\psi = \partial _{0}\psi -\nabla\psi$ and then took the conjugate $\partial_\mu\psi^{*} = \partial _{0}\psi^{*} -\nabla\psi^{*}$ and then multiply by $\partial^\mu\psi = \partial^{0}\psi -\nabla\psi$ but at the end I get that the canonical momentum is:
$\partial _{0}\psi^{*} -\nabla\psi^{*}$
what is wrong?