Let $f,g \colon U \longrightarrow \mathbb{C}$ be two holomorphic functions on a small open set $0 \in U\subset \mathbb{C}^n$ (or even germs at the origin) such that $f(0) = g(0) =0$. Suppose that the singular sets of the $1$-forms $\mbox{d}f$ and $\mbox{d}g$ have codimension at least two. If $$\mbox{d}f \wedge \mbox{d}g = 0,$$ is it possible to write $g$ in terms of $f$? This gives relations between the derivatives of $f$ and those of $g$. However, I was not able to figure out how to relate $g$ and $f$.
From the de Rham-Saito division lemma one can write $$\mbox{d}g = h \mbox{d}f,$$ for some nonvanishing holomorphic function $h$. Could someone provide an example where $h$ is not constant?