Assume that $U$ is an open subset of $\mathbb{C}^n$. We fix the standard almost complex structure $J$ on $U$. Assume that $\omega$ is a symplectic structure on $U$.

Is there a Riemannian metric on $U$ with the following property?

For every smooth function $f: U \to \mathbb{R}$, we have $J \nabla f = X_f$ where $X_f$ is the Hamiltonian vector field corresponding to $f$

Assume that $U$ is an open subset of $\mathbb{C}^n$. We fix the standard almost complex structure $J$ on $U$. Assume that $\omega$ is an arbitrary symplectic structure on $U$.

Is there a Riemannian metric on $U$ with the following property?

For every smooth function $f: U \to \mathbb{R}$, we have $J \nabla f = X_f$ where $X_f$ is the Hamiltonian vector field corresponding to $f$