The Sasaki Complex Structure
This is my attempt to depict the complex structure associated to the Sasaki metric on the positive octant of the sphere. As the Sasaki metric is inherently 4 dimensional whereas this image is 3 dimensional, the fact that X and -JX are perpendicular is definitely lost in translation. Note that the blue points are in the sphere, the red points are in the red plane and the green point is in the green plane. Points B,V, and X can all be manipulated, which will affect B' and -JX.
For computational ease and to make the picture clearer, I took a few artistic licences. Firstly, to represent tangent vectors to the tangent bundle (i.e. elements of TTM), I used arrows between two points in the tangent bundle, which hopefully got the idea across. With that being done, I used a first order Taylor series instead of trying to integrate out the exponential of X to compute B'. More importantly, in order to convey that the complex structure does not change the fiber direction, I orthogonally projected the vector from B to V into the tangent space of B' (i.e. from the red tangent space to the green) and then renormalized so that the lengths were preserved. As changing the base point stretches and skews the fibers, this isn't the most geometrically faithful representation, but is easier to picture.