The awkward title is an attempt at approximating the following specific question: Let $(M^{2n}, J)$ be a complex manifold, suppose $g_0$ is a Riemannian metric $M$ compatible with $J$, and suppose $\phi_t$ is a one parameter family of diffeomorphisms of $M$ generated by a vector field $X$ such that $\phi_t^* g$ is compatible with $J$ for all $t$. The questions is: must $X$ be a (real part of a) holomorphic vector field in the sense that $L_X J = 0$?

I can give a proof that yes, it is true, in the case that $g$ is also Kahler, by showing that one has that the family of Kahler forms $\omega_t = (\phi_t^* g)(\cdot, J \cdot)$ satisfies $\omega_t = \phi_t^* \omega_0$, after which one obtains $\phi_t^* J = J$, and the claim follows. In general the hypotheses imply that $L_X g$ is of type $(1,1)$, and the claim seems equivalent to showing that $L_X \omega$ is also of type $(1,1)$, but I can only show this with the Kahler hypothesis.