Let $X$ be a compact oriented Riemannian manifold. By Hodge decomposition, we can decompose
$$\Omega^k(X)=\mathrm{im}(d)\oplus\mathrm{im}(d^*)\oplus\ker(\Delta).$$
Now, if further $X$ has a complex structure, I read on a book that
$$\Omega^{p,q}(X)=\partial(\Omega^{p-1,q}(X))\oplus\partial^*(\Omega^{p+1,q}(X))\oplus\ker(\Delta_{\partial}),$$ or $$\Omega^{p,q}(X)=\bar\partial(\Omega^{p,q-1}(X))\oplus\bar\partial^*(\Omega^{p,q+1}(X))\oplus\ker(\Delta_{\bar\partial}).$$
My question is: given that I know the theorem for the real case and $d=\partial+\bar\partial$, shouldn't the result be
$$\Omega^{p,q}(X)=(\partial(\Omega^{p-1,q}(X))+\bar\partial(\Omega^{p,q-1}(X)))\oplus(\partial^*(\Omega^{p+1,q}(X))+\bar\partial^*(\Omega^{p,q+1}(X)))\oplus\ker(\Delta)?$$