The laplacian of a complex function

Question

For a complex function $f$, we know its Laplacian is $4 \frac{\partial}{\partial z}\frac{\partial}{\partial \bar z}$. But now I am trying to calculate $\nabla |f|^2$. I am looking for a formula which would help me simplify my calculation. Something possibly of the form $\nabla \langle f, f \rangle = \langle \nabla f, f \rangle + \langle f, \nabla f \rangle$. Maybe there are extra terms invovling the gradient also.