For a holomorphic $f:\mathbb{C}^n\rightarrow \mathbb{C}$ and $a=(a_1, \dots, a_n) \in \mathbb{C}^n$, let $f_a:\mathbb{C}^n\rightarrow \mathbb{C}$ be the function $$(z_1, \dots, z_n) \mapsto a_1z_1 + \dots + a_nz_n + f$$
For general $a$, does $f_a$ have only nondegenerate critical points?
(Nondegeneracy is defined here in terms of the holomorphic Hessian bilinear form.)
This is true (at least locally) if $g=\text{grad}(f)$ has isolated zeros. A proof can be found in Appendix B of Milnor's "Singular points of complex hypersurfaces".
The idea is to relate the rank of the Jacobian of $g$ (=Hessian of $f$) to the local degree of the mapping $g: \mathbb{C}^n \rightarrow \mathbb{C}^n$.