$f$ can only be quadratic if its Hessian determinant is constant. How about complex case

Question

It is known that for $f:\mathbb{R}^{n}\rightarrow\mathbb{R}$, if $f$ is convex, then $f$ can only be quadratic if the Hessian determinant $\det\Big(\frac{\partial^{2}f}{\partial x_{i}\partial x_{j}}\Big)$ is constant. Can the result generalize to the following? If $f:\mathbb{C}^{n}\rightarrow\mathbb{C}$ is analytic, then if $\det\Big(\frac{\partial^{2}f}{\partial z_{i}\partial z_{j}}\Big)$ is constant, $f$ can only be quadratic in $z_{i}$, where $z_{i}$ are the complex variable. What will be the conditions so that the above is true?