Set-up: Let $(L,g)$ be a real-analytic riemannian manifold. And let $T^*L$ denote the cotangent bundle of $L$.
According to Bruhat-Whitney, on a suffciently small neigbourhood of the zero section in $T^*L$ there exists a complex structure $J$ which is unique up to biholomorphism.
The zero section $L \subset T^*L$ is Lagrangian and thus totally real. By this answer real analytic and totally real means we can find local holomorphic coordinates $\{z_i = x_i +i y_i\}_{i=1}^n$ such that $L = \{y =0\}$.
Question: Does it follow that in these coordinates the complex structure $J$ on this neighborhood of $L$ in $T^*L$ is given by $$ J\frac{\partial}{\partial x_i}= \frac{\partial}{\partial y_i}, \quad J\frac{\partial}{\partial y_i}= -\frac{\partial}{\partial x_i}? $$