This is found at the bottom of page 9 of David Tong's QFT lectures. The Euler-Lagrange equations for the complex scalar field:
$$\mathcal L=\frac{i}{2}(\psi^*\dot\psi-\dot{\psi^*}\psi)-\nabla\psi^*\cdot\nabla\psi-m\psi^*\psi \tag{1.15}.$$
However, to obtain the equation of motion for $\psi^*$, we need the following derivatives of the Lagrangian density:
$$\partial_\mu\left(\frac{\partial\mathcal L}{\partial(\partial_\mu\psi^*)}\right),\quad \frac{\partial\mathcal L}{\partial\psi^*}.$$
In the notes, the following derivatives are instead used:
$$\frac{\partial\mathcal L}{\partial\psi^*},\quad \frac{\partial\mathcal L}{\partial\dot{\psi^*}}, \quad \frac{\partial\mathcal L}{\partial\nabla\psi^*}. \tag{1.16}$$
I don't see why we can use these instead (ie. take the derivative of $\mathcal L$ wrt. the time derivative and the gradient of $\psi^*$ separately and then combine them, which I think is what's being done here).