Complex structure on cotangent bundle If $M$ is a complex manifold with complex structure $J$, why does the cotangent bundle of $M$ carry a natural complex structure, and not an almost complex structure. Is that obvious?

As Olivier suggests, you can define a complex structure on $T^*M$ directly using the complex structure on $M$. This is basically the complex version of Proposition $3.18$ of Lee's _Introduction to Smooth Manifolds_ (second edition) except that instead of $T^*M$, Lee is using $TM$. If you would like further information on this approach, let me know.