Suppose $u$ is a real-valued $C^2$ function on $\mathbb{C}^n$, I wonder if the determinant of $$ \left( \dfrac{\partial u}{\partial z^i \partial \bar{z}^j} \right)_{i\bar{j}} $$ is related to the determinant of $$ \begin{pmatrix} \left( \dfrac{\partial u}{\partial x^i \partial x^j} \right)_{ij} & \left( \dfrac{\partial u}{\partial x^i \partial y^j} \right)_{ij} \\ \left( \dfrac{\partial u}{\partial y^i \partial x^j} \right)_{ij} & \left( \dfrac{\partial u}{\partial y^i \partial y^j} \right)_{ij} \end{pmatrix} $$
If they were, when will they coincide?
If $n=1$, $$\frac{\partial u}{\partial \overline{z}} = \frac{\partial u}{\partial x} + i \frac{\partial u}{\partial y}.$$
Therefore, $$\frac{\partial^2 u}{\partial z \partial \overline{z}} = \frac{\partial u}{\partial x \partial x} + i \frac{\partial^2 u}{\partial x \partial y} -i \frac{\partial^2 u}{\partial x \partial y} + \frac{\partial^2 u}{\partial y \partial y} = \frac{\partial^2 u}{\partial x \partial x} + \frac{\partial^2 u}{\partial y \partial y}.$$
The determinant of the real Hessian is $$\frac{\partial^2 u}{\partial x \partial x} \cdot \frac{\partial^2 u}{\partial y \partial y} - \left( \frac{\partial^2u }{\partial x \partial y } \right)^2.$$