Suppose that $f \colon {\mathbb{C}}^n \to \mathbb{C}$ be twice differentiable and let \begin{equation*} \frac{\partial}{\partial z_k} := \frac{1}{2}\left( \frac{\partial}{\partial x_k} - i \frac{\partial}{\partial y_k} \right) \quad \text{ and } \quad \frac{\partial}{\partial \bar{z}_k} := \frac{1}{2}\left( \frac{\partial}{\partial x_k} + i \frac{\partial}{\partial y_k} \right) . \end{equation*} Then the complex Hessian is the matrix \begin{equation*} \begin{bmatrix} \frac{\partial^2 f}{\partial z_1 \partial \bar{z}_1} & \frac{\partial^2 f}{\partial z_1 \partial \bar{z}_2} & \ldots & \frac{\partial^2 f}{\partial z_1 \partial \bar{z}_n} \\ \frac{\partial^2 f}{\partial z_2 \partial \bar{z}_1} & \frac{\partial^2 f}{\partial z_2 \partial \bar{z}_2} & \ldots & \frac{\partial^2 f}{\partial z_2 \partial \bar{z}_n} \\ \vdots & \vdots & \ddots & \vdots \\ \frac{\partial^2 f}{\partial z_n \partial \bar{z}_1} & \frac{\partial^2 f}{\partial z_n \partial \bar{z}_2} & \ldots & \frac{\partial^2 f}{\partial z_n \partial \bar{z}_n} \end{bmatrix} . \end{equation*} When applied to tangent vectors of the zero set of $f$ , it is called the Levi form and used to define a Levi pseudoconvex point of a boundary of a domain. Note that the complex Hessian matrix is not the same as the normal (real) Hessian. A twice continuously differentiable real valued function with a positive semidefinite real Hessian matrix at every point is convex, but a function with positive semidefinite complex Hessian matrix at every point is plurisubharmonic (since it's continuous it's also called a pseudoconvex function).

Bibliography

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Steven G. Krantz. Function Theory of Several Complex Variables, AMS Chelsea Publishing, Providence, Rhode Island, 1992.