@bsaucer: In one sense the hairy ball theorem is still true for a field of complex tangent vectors to the ordinary sphere. I.e. one constricts the real tangent space to the real sphere by gluing copies of R^2 together using the the derivatives of the smooth coordinate maps. Then it makes sense to speak of smooth tangent vector fields, and any smooth tangent vector field on the sphere must have 2 zeroes, "counted properly". But one can also cover the sphere by open sets whose coordinate maps have not just smooth but holomorphic compositions, hence one can also construct the sphere as a complex holomorphic manifold of complex dimension one, and by gluing copies of C ≈ R^2 together by the derivatives of these holomorphic coordinate maps, one gets the "same" object, i.e. the sphere with a real 2 dimensional vector bundle over it, only this time the fibers of the vector bundle are copies of C, i.e. R^2 with more structure.
Now a holomorphic complex tangent vector field on the complex Riemann sphere, is in particular also a smooth tangent vector field on the underlying real sphere, hence the holomorphic complex hairy ball theorem is implied by the smooth one.
In particular I am interpreting a "covering" of the complex sphere by a family of complex vectors, as a section of the complex tangent bundle, i.e. one vector at each point of the sphere. So here the real smooth 2-diml sphere, and its rank 2 real tangent bundle, have been given the structure of a 1-diml holomorphic complex manifold, with a rank one complex tangent bundle. In relation to jbergman's answer, we have added to the real tangent bundle, a rotation operator on each planar fiber, i.e. a real linear operator J with J^2 = -Id, varying smoothly over the sphere. Such an operator is called an "almost complex structure", on the real sphere. (Having an almost complex structure on the tangent bundle is a necessary, but not sufficient, condition for the real smooth manifold itself to have a complex holomorphic structure. In particular such an almost complex structure already forces the real manifold to have even real dimension, since the only irreducible factor of the minimal polynomial, hence also of the characteristic polynomial, is X^2 +1.)
You can check this result by computing the poles of the one - form dz, at infinity on the Riemann sphere. I.e. a one - form is a section of the dual, or cotangent, bundle, hence should have degree -2 instead of 2 for the tangent bundle, i.e. 2 poles instead of 2 zeroes. In fact dz has no zeroes or poles in the finite plane, and using the coordinate w = 1/z at infinity, we have zw = 1, so zdw + wdz = 0, so dz = -zdw/w = -dw/w^2, a pole of order 2 at w=0.
It is actually enough to check the result for a single one-form, since dividing two one forms defines a meromorphic function, i.e. a holomorphic map to the Riemann sphere, which sends the same number of points to zero as to infinity, i.e. such a map has a well defined degree d. Hence for both forms, #zeroes - #poles = the same number, (which is just d if both forms are holomorphic with distinct zeroes).
The hairy ball theorem for a complex holomorphic Riemann surface of genus g, says that a holomorphic 1-form has 2g-2 zeroes, counted properly, or for a meromorphic form, #zeroes - #poles = 2g-2. The nice part in the holomorphic or meromorphic case, is there are always only a finite number of zeroes and poles, at least for a form that is not identically zero, and their order can be computed from a Taylor series expansion.
In answer to your original question, complex vectors look exactly like real vectors. As jbergman said, a complex vector space is just an even dimensional real vector space plus a real linear rotation operator T. The vectors don't look any different, the only role of the complex structure is that you are given a way to multiply the real vectors not only by real scalars but also by i, i.e. to rotate each one "90 degrees" in a certain real 2 dimensional subspace. I.e. given a real linear operator T with T^2 = -Id, and hence minimal polynomial X^2 + 1 = 0, on a real vector space V, we get an action on V of the field k[X]/(X^2+1) ≈ C, i.e. a structure of complex vector space on V. Any basis for V over this field decomposes V into a direct sum of real 2 dimensional T invariant subspaces, i.e. "complex" subspaces.
