(page 17)
...It follows that the projection (of tangent spaces, respectively real, complexified and holomorphic, at $p$ to a complex manifold $M$)
T_{\mathbb{R},p}(M)\longrightarrow T_{\mathbb{C},p}(M)\longrightarrow T_{p}'(M)
is an $\mathbb{R}$-linear isomorphism. This last feature allows us to "do geometry" purely in the holomorphic tangent space. For example, let $z(t) = x(t) + iy(t)$, and the tangent to the arc may be taken either as x'(t)\frac{\partial}{\partial x}+ y'(t)\frac{\partial}{\partial y} in $T_{\mathbb{R},p}(\mathbb{C})$ or
z'(t)\frac{\partial}{\partial z} in T'(\mathbb{C}) and these two correspond under the projection.
What does the book mean by "doing geometry"?
I guess more basically, what properties precisely do $\mathbb{R}$-linear isomorphisms preserve that are fundamental to 'doing geometry'? I assume only angles?