It asks for a generalization of the question in the post

Normal form for a holomorphic Morse function

Suppose $f$ is a holomorphic function on a complex manifold $M$ which has Bott type critical points, i.e., $df$ vanishes along a complex submanifold $S\subset M$ with nondegenerate Hessian in the normal direction to $S$. For each $p \in S$, can we find a holomorphic local coordinates $z_1, \ldots, z_s, z_{s+1}, \ldots, z_m$ such that locally

$$f(z_1, \ldots, z_m) = \sum_{j = s+1}^m z_j^2 + C,$$ where $s = {\rm dim} S$?