Let M be a complex manifold with real tangent bundle TM. Let $J$ be its associated almost complex structure($JoJ=-id$) and $\nabla$  a torsion free, flat connection in $TM$ compatible with $J$, that is $$\nabla J=0.$$
 Is it true that in this case the manifold is geodesically complete?


If one replaces the complex structure by a simplectic structure  the question becomes a simpler version of Markus conjecture.