I have a naive question about real coordinates of a complex manifold. Let's consider 1-dimensional case for simplicity. Let $X$ be a Riemann surface and $z$ be a local complex coordinate. Then one often claims that $z$ and $\bar z$ are real coordinates of the underlying real surface. Why is this true?
I think $z$ and $\bar z$ are related by the complex conjugate and doesn't seem independent. Also it seems to me that $z$ already move in $2$-dimensional direction. Could anyone clarify what is going on?
No, $z$ and $\bar z$ are not in any sense real coordinates on $X$. As @Travis noted, the real and imaginary parts of $z$ are real coordinates, but that's not the same thing.
There is an interesting way to "pretend" that $z$ and $\bar z$ are independent coordinates, when thinking about holomorphic functions. Suppose $f\colon \mathbb C\to \mathbb C$ is a real-analytic function, meaning that in a neighborhood of each point, it has a convergent power series expansion in $(x,y)$ coordinates (where $z=x+iy$). By making the substitutions $x=\text{Re} \,z= \tfrac 1 2 (z+\bar z)$ and $y = \text{Im} \,z = \tfrac 1 {2i} (z - \bar z)$, we can rewrite $f$ as a convergent series in powers of $z$ and $\bar z$. Now formally substitute a new complex variable $w$ in place of $\bar z$, and you get a power series in two complex variables $\widetilde f(z,w)$. It turns out that your original function $f$ is holomorphic in $z$ if and only if the new function $\widetilde f(z,w)$ is independent of $w$. Informally, we say in this case that $f$ is "independent of $\bar z$."