If $X$ is a Riemann surface, does $T_p^*X$ have complex dimension $1$ or $2$? $X$ is 1-dimensional complex manifold but why does $T_p^*X$ have dimension 2 over $\mathbb{C}$? 
Shouldn't a tangent space of an $n$-dimensional manifold over a field $F$ have dimension $n$ over $F$? 

Forster means the complexified cotangent bundle of $X$, that is the complex vector bundle obtained by $\left(T_pX\right)^{\vee}\otimes_{\mathbb{R}}\mathbb{C}$; its complex dimension at any point is $2$.
UPDATE: Some book.


*

*Daniel Huybrechts - Complex Geometry. An Introduction, chapter 2, section 1;

*Raymond O. Wells, Jr., Oscar Garcia-Prada - Differential Analysis on Complex Manifolds, chapter 1, section 3.

