Background: Let $(x,y)= (x^1, \dots , x^n, y_1, \dots , y_n)$ be a canonical coordinate chart on $T^*M$ about some point $q=(p,0)\in T^*M$.
i.e., given coordinates $x =(x^i)$ on $M$ about $p$, $$(x^1, \dots , x^n, y_1, \dots , y_n) \longrightarrow (x^1, \dots , x^n, y_1 dx^1, \dots , y_ndx^n).$$
Questions: In what instances are $z= (z^i) = (x^i + \sqrt{-1} y_i)$ complex coordinates on $T^*M$ (i.e., are there conditions we can place on $M$?)
Thoughts: We want the transition functions to be holomorphic functions between open sets in $C^n$. However the change of coordinates on $T^*M$ are completely determined by the coordinates on $M$. So I am thinking maybe we can restrict to coordinates $x$ and $\tilde{x}$ on $M$ such that the transition functions satisfy some property.
Let $(\phi, U)$ and $(\tilde{\phi}, \tilde{U})$ be charts on M. Then $(\phi \times Id, U\times \mathbb R^n)$ and $(\tilde{\phi} \times Id, \tilde{U}\times \mathbb R^n)$ are charts on $T^*M$. So we want to show $$F =(\phi \times Id) \circ (\tilde{\phi} \times Id)^{-1}= \phi \circ \tilde{\phi}^{-1} \times Id: W \times \sqrt{-1} \mathbb R^n \rightarrow \tilde{W} \times \sqrt{-1} \mathbb R^n,$$ where $W = \phi^{-1} \left( \phi(U) \cap \tilde{\phi}(\tilde{U})\right)$ and $\tilde{W}$ is defined analagously. So this becomes a condition on the real part of $F$ which is the transition function on $M$. Would maybe something like pluriharmonic work? And would this even make sense for a transition function?
Since cotangent and tangent bundle are isomorphic, could we possibly phrase this condition on $M$ in terms of $G$-structures or a reduction of the structure group?
A very nice, well explained, intriguing question. I wish I knew the answer. My preliminary guess is pluriharmonic as explained in Maciej Klimek's London Mathematical Society Monograph "Pluripotential Theory" (Oxford: 1991) may help. Tangent space is discussed starting on page 245. Sorry if I am giving you a bum steer.