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.
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 $X$.
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.
