If M is a complex manifold with complex structure J, why the cotangent bundle of M carries 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 Lemma 4.1 of Lee's Introduction to Smooth Manifolds except that instead of $T^*M$, Lee is using $TM$. If you would like further information on this approach, let me know.