Can a non-Kähler complex manifold be rationally connected? Let $X$ be a compact complex manifold. Suppose that $X$ is rationally connected in the sense that any two points lie in the image of a rational curve $\mathbb{CP}^1 \to X$. Are there any non-Kähler examples of such $X$?
If $X$ is Kähler and rationally connected, then $X$ is projective. So I might suspect that being rationally connected may force some algebraic structure on $X$, but I cannot find any reference for this question.
 A: I am writing my comment as a question.  I have certainly explained these examples before on MathOverflow, since they show that the Kollár-Miyaoka-Mori conjecture cannot hold beyond Fujiki class $\mathcal{C}$ (roughly, in the setting of Kaehler manifolds).
Let $C$ be a copy of $\mathbb{CP}^1$.  Let $E$ be a (geometric) holomorphic vector bundle over $C$ of rank $2$ that is ample.  Denote by $E^*$ the open complement in $E$ of the zero section.  Consider the holomorphic, fiberwise, linear action of the discrete group $\mathbb{Z}$ on $E^*$ by scaling by $2$ (or by any invertible complex number with modulus different from $1$).  This action is free.  Denote the quotient by $$\pi:E^*\to X.$$  This morphism factors the projection from $E^*$ to $C$, so there is an induced proper, holomorphic submersion, $$\rho:X\to C.$$
The compact complex manifold $X$ is not in Fujiki class $\mathcal{C}$ since the first Betti number equals $1$.  Yet it is rationally connected in the sense that any two points are contained in the image of a holomorphic map from $\mathbb{CP}^1$.
Indeed, for any two points of $C$ (possibly the same point twice) and for any two elements of $E$ in the fibers over these points, there is a high-degree self-map $\mathbb{CP}^1\to C$ and distinct points of $\mathbb{CP}^1$ lying over the two points of $C$ (possibly the same point, which means the two distinct points are distinct preimages of this one point of $C$) such that the pullback of $E$ to $\mathbb{CP}^1$ is "very, very ample".  Thus, there exists a global section of the pullback that is everywhere nonzero, and that has the specified values over the two distinct points of $\mathbb{CP}^1$.  This global section defines a holomorphic map from $\mathbb{CP}^1$ to $E^*$ that connects the two specified points of $E^*$.  The composition with $\pi$ is the desired holomorphic map to $X$.
A: As Jason said already, there are many
examples of Moishezon manifolds which
are rationally connected. Indeed, any
manifold bimeromorphic to a rational
connected manifold is again rationally
connected, and there are many non-Kahler
Moishezon manifolds bimeromorphic to
$CP^n$ (or any  other Fano manifold,
I suppose).
The better question would be whether all
rational connected manifolds are Moishezon.
The answer is, suprisingly, positive,
if you define rational connectedness
one way, and negative for another definition.
Recall that a rational curve is called
ample if its normal bundle
is $\bigoplus O(i_k)$ with all $i_k$ positive.
The first definition is
Defintion:
A compact complex manifold $M$ is called 
rationally connected if there is an ample
curve with compact deformation space (Barlet
or Douady, does not matter).
In this case, any rationally connected manifold
is Moishezon, as shown by Campana
F. Campana, Reduction algebrique d'un morphisme
faiblement Kahlerien propre, Math. Ann. 256 (1980), 157--189.
Another definition is the same, but without the compactness
assumption. In this case you have manifolds which
are rationally connected, but not Moishezon, for example,
the twistor space of K3 surface or a torus. Curiously,
there is a rational curve passing through  every
$n$ points of such a manifold; this is explained,
for example, in my paper 
"Rational curves and special metrics on twistor spaces".
Personally, I think the first definition makes more sense.
