Reading Forster's Lectures on Riemann surfaces I find his definition of holomorphic differential forms unpleasant. Let $X$ be a Riemann surface. He says that they are forms $\omega$ with value in $T^{1,0}$ (the complex line generated by $\mathrm{d}z$ in local coordinates, or equivalently the eigenbundle of the complex cotangent space $T^*X \otimes \mathbb{C}$ associated to the eigenvalue $i$ of the section $J$ of $\mathrm{End}(T^*X \otimes \mathbb{C}) $ obtained from the complex structure on the real tangent space), such that in local coordinates given by $z$, the function $f$ such that $\omega=f\mathrm{d}z$ be holomorphic on $X$, which is the same as $\omega$ being closed. Now, for instance, for a real $C^k$ manifold $M$, a $C^k$ $p$-form $\omega$ is any section of $\Lambda^p T^*M$ (that is any function of sets $M \to \Lambda^p T^*M$ that maps $\{a\}$ to $T_aM$) such that for any $p$ $C^k$-vector fields $X_1,\dots,X_p$, the resulting function $\omega(X_1,\dots,X_p)$ is $C^k$. Is there an analogous characterization of holomorphic differential forms, perhaps with a complex manifold structure on $T^*X$ ?
Definition of holomorphic differential forms
complex-geometryriemann-surfacesstochastic-differential-equations
Related Solutions
A complex differential form can be thought of as two real differential forms - the real and imaginary parts, just like complex numbers. As you say, the exterior derivative is then the linear extension of the real exterior derivative. Namely,$$d(\alpha+i\beta)=d\alpha+id\beta.$$If you want a rigorous proof for the isomorphism mentioned in the last paragraph, you can do it locally, as Mike comments. This is the standard way to handle vector bundles.
On the one hand what you're suggesting is rather easy to do. Take a complex manifold $X$ of complex dimension $n$ (so real dimension $2n$). The set of "things that look like $f dz_1 \wedge \cdots \wedge dz_n$ locally" is a very important object, called the canonical line bundle of $X$ and denoted $K_X$. As Didier noted, the things in $K_X$ ("the sections of $K_X$") are differential $n$-forms and can be integrated over submanifolds of $X$ of real dimension $n$.
Unfortunately this is not a super interesting thing to do if you're a complex geometer, because if $n$ is even and you can find a complex submanifold of (real) dimension $n$ in $X$, the integral of anything in $K_X$ over it will be zero. The reason is essentially one of linear algebra, but vaguely speaking to get a nonzero integral over a complex manifold you need the same number of $dz$ and $d\bar z$ in your form, and we only have $dz$'s here.
"But wait," you say, "I integrate holomorphic forms like $f dz$ in the plane over paths all the time and I love it."
That is true, but turns out to be a rather specific feature of complex dimension $1$. There a form like $f dz$ can be integrated over a real path, and closed such paths are the boundaries of open complex submanifolds, so Stokes' theorem applies. By Cauchy's theorem, the real money in integrating holomorphic forms is really in integrating meromorphic forms, that is forms with poles. In higher dimensions the poles of a meromorphic form are no longer isolated and can't be contained inside the boundary of a real hypersurface (and even less inside a lower-dimensional real manifold).
What we can do in higher dimensions is to take two holomorphic $n$-forms $\alpha$ and $\beta$ and try to integrate $\alpha \wedge \overline{\beta}$ over $X$, as that will be a $2n$-form on a (real) $2n$-dimensional manifold. This leads to things like Bergman kernels and Weil-Peterson metrics, which are both active areas of research. There is also some kind of theory of residues in higher dimensions, but it is less developed and important than in complex dimension one.
Best Answer
The analogous definition you are probably looking for is just as holomorphic sections of $\bigwedge^p T^*X$ (where a holomorphic section is the same thing as a $C^k$ section where we just require the function in question to be holomorphic instead). This definition is just fine and we usually denote these sections $H^0(X, \bigwedge^p T^*X)$. Note that as a matter of fact $ \bigwedge^p T^*X \cong \bigwedge^{p,0} T^*X := \bigwedge^p \left( T^{1,0}\right) ^*X$ as complex vector bundles, so the definitions are indeed equivalent.
You have to be careful when you say "which is the same as ω being closed" thought. This is not true, the form $\bar{f} d\bar{z}$ is closed (On a Riemann surface!) just as well. In general we denote with $\mathcal{A}^{p,q}(X)$ the $smooth$, i.e. $C^\infty$-sections of form $\sum f_I \hspace{0.1cm} dz^{i_1}\wedge ...\wedge dz^{i_p} \wedge d\bar{z}^{j_1} \wedge ... \wedge d\bar{z}^{j_1}$. In this case the elements of $\mathcal{A}^{p,0}(X)$ that are closed under the action of the Dolbeaut-operator $\bar{\partial}$ are precisly the holomorphic forms $H^0(X, \bigwedge^p T^*X)$ and I guess that is what you meant when you said "closed".
Now, I guess the real thing you wanted to know is why we have this asymmetry in the definition of holomorphic forms. When we talk about $C^k$-manifolds we consider $C^k$-sections, so why do we care so much about the smooth sections in the complex case? One answer is, that differential forms can be used to compute the cohomology of manifolds (roughly the shape - how many holes?). The thing is that smooth functions are in some sense way easier to handle since they admit partitions of unity, while holomorphic functions that are determined on an ever so small open set are already uniquely determined everywhere they can be defined. For this reason we can e.g smoothly embbed every $C^\infty$-manifold in $\mathbb{R}^n$ (since we can do so locally), but we can not embbed in a holomorphic fashion every complex manifold into $\mathbb{C}^n$. Similarly in the real case we have the beautiful theory of de Rham which tells us how to compute the cohomology of smooth manifolds. Due to the difference between smooth and holomorphic functions we can not do exactly the same thing in the complex scenario but we should not be afraid to "downgrade" and use only the smooth structure to calculate cohomology which in turn will tell us interesting things about our manifold.