Meaning of the factor $\mathbb{C}$ in the definition of a complex valued 1-form? A complex valued 1-form $\alpha$ is defined as $\alpha \in C^{\infty}(T^*X\otimes\mathbb{C}$); that is if I understand it correctly a smooth function on the tensor product of the cotangent bundle with $\mathbb{C}$.
What is the factor $\mathbb{C}$ needed, what is it doing in that definition?
And while we are at it, why is this 1-form defined on the cotangent bundle?
Naively imagining 1-forms as some kind of co-vectors, I would rather have expected it do be defined on the tangent bundle $TX$ ...
 A: The cotangent space at each point is a real vector space, and tensoring it with $C$ turns it into a complex vector space with the same dimension, corresponding to taking a basis and allowing the real coefficients to be complex. 
1-forms are the sections on cotangent space, so complex-valued 1-forms are the sections of the complexified cotangent space.
A: When we write $C^\infty(T^*M\otimes\mathbb{C})$ in this context, that doesn't mean we have functions defined on $T^*M\otimes\mathbb{C}$.  Rather, it means we mean certain functions which take values in $T^*M\otimes\mathbb{C}$, specifically smooth sections of the projection map $T^*M\otimes\mathbb{C}\to M$.  More generally, if $E$ is a vector bundle on $M$, it is common to write $C^\infty(E)$ for the space of smooth sections of $E$.  (This is unfortunately ambiguous with the notation $C^\infty(M)$ for the ring of smooth functions on $M$.  A better notation is $C^\infty(M,E)$ for the space of sections of $E$ on $M$, but people sometimes abbreviate this to just $C^\infty(E)$.)
So that is why we have the cotangent bundle--at each point, we are picking a cotangent vector.  The factor of $\mathbb{C}$ is because the cotangent space $T^*M_p$ at a point $p$ is the space of real linear functionals on the tangent space: they take in a tangent vector, and spit out a real number.  By tensoring with $\mathbb{C}$, you get (up to isomorphism) the space of complex linear functionals on the tangent space, which spit out complex numbers instead.  More generally, if $V$ is a real vector space, there is a natural isomorphism $\operatorname{Hom}(V,\mathbb{R})\otimes\mathbb{C}\to\operatorname{Hom}(V,\mathbb{C})$ (induced by the bilinear map which sends a pair $(f,c)\in(\operatorname{Hom}(V,\mathbb{R}),\mathbb{C})$ to the linear map $g(v)=cf(v)$).
So to sum up, a section of $T^*M\otimes\mathbb{C}$ is a function that at each point of $M$ gives you a complex-valued linear functional on the tangent space.  That's a complex-valued 1-form.
