Identification of the holomorphic tangent space with the real tangent space

Reviving an old post here.

As I am sure Griffiths and Harris mention in the book, the bundle $T'M$ (or $T^{1,0}M$ in other books) is supposed to be the holomorphic tangent bundle. One way to specify what that is to say that it should act on holomorphic functions, in other words it should consist of derivations mapping holomorphic functions back to holomorphic functions.

Now note that given a holomorphic function $f$ on an open subset of $\mathbb{C}^n$, we have $$ \frac{\partial{f}}{\partial{x_j}}=\frac{\partial{f}}{\partial{z_j}}, \quad \frac{\partial{f}}{\partial{y_j}}=i\frac{\partial{f}}{\partial{z_j}}$$where $\frac{\partial{}}{\partial z_j}$ can either be given by the formula you mention in your post, or simply as the partial differentiation symbol in the $j$-th direction (which makes sence, because the limit in the partial derivative exists by the virtue of $f$ being holomorphic). In this sense it is a pretty natural identification.