A complex polynomial with partial derivatives equal to zero is constant.
If $l\geq 1$ or $m\geq 1$, since we applied at least one time $\frac{\partial f}{\partial z}$ or $\frac{\partial f}{\partial \bar z}$, $\frac{\partial^{l+m}}{\partial z^m\partial \bar z^l}p$ must be equal to $0$. So $l!m!a_{lm}=0$ and since $l!m!\neq 0$ we have $a_{lm}=0$ if $l\geq 1$ or $m \geq 1$. It shows that the only potentially non-vanishing term is $a_{0,0}$.