Is this relation between Wirtinger operators and gradient correct?
$\def\p{\partial} \def\pc{\overline{\partial}}$We have \begin{align*} \nabla f\cdot\nabla g &= \p_x f\p_x g+\p_y f\p_y g\\ &= (\p f+\pc f)(\p g+\pc g) + i(\p f-\pc f)i(\p g-\pc g) \\ &= \p f\p g + \p f \pc g + \pc f \p g + \pc f\pc g \\ &\quad -(\p f\p g - \p f \pc g - \pc f \p g + \pc f\pc g) \\ &= 2(\p f\pc g + \pc f\p g), \end{align*} as claimed. This comes directly from the definition of the Wirtinger derivative.