Counterexample to decomposition of Harmonic Forms

Question

It is well known that if $X$ is a Kahler manifold, then $$\bigoplus_{p+q=k}\mathcal{H}^{p,q}=\mathcal{H}^{k}(X,g)_{\mathbb{C}}=\mathcal{H}^{k}_{\overline{\partial}}(X,g)=\mathcal{H}_{\partial}^{k}(X,g)$$ If one loosens the restriction that $X$ is Kahler to simply being Hermitian, are there any quick counter examples to show that these decompositions do not coincide?