What is a good geometric way of thinking of complex tangent vectors on a manifold? I can convince myself that I understand tangent vectors by thinking of them as paths on the manifold. Is there a nice way to visualize or think of complex vectors on a manifold?
I mean, I know the definition of a complexified tangent bundle, and if a manifold has an almost complex structure, and I know what it means for the complex vector to be holomorphic, antiholomorphic, and that there is an isomorphism between these eigenspaces and the real tangent bundle...
But I just don't "get" it. I feel like I have no intuitive understanding of what these things are.
If the manifold comes with an almost complex structure, is it correct to think of holomorphic tangent vectors as some kind of germs of J-holomorphic maps from an open set in $\mathbb{C}$ into the manifold?