A toy SYZ mirror symmetry is described as follows. Let $X$ be a $n$-dimensional compact manifold. The contangent bundle $T^*X$ has a natural symplectic structure locally given by the $2$-form $\sum_idq_i \wedge dx_i$. Here $(x_i)$ is a local coordinate of $X$ and $(x_i,q_i)$ is a local coordinate of $T^*X$ corresponding to the $1$-form $\sum_i q_i dx_i$.

On the other hand, the tangent bundle $TX$ has a complex structure locally given by $(x_i+iy_i)$. Here $(x_i)$ is a local coordinate of $X$ and $(x_i,y_i)$ is a local coordinate of $TX$ corresponding to the vector field $\sum_i y_i \partial/\partial x_i$.

Some people want to compactify the two spaces by modding out the fibers by a lattice and its dual. Then the local holomorphic coordinate becomes $z_i=\exp2\pi \sqrt{-1}(x_i+iy_i)$. This is a toy SYZ mirror symmetry I am aware of.

My question is, why the complex structure on $TX$ is natural? Alsoare there any more good examples that shows the SYZ mirror symmetry?