There are various notions of 'harmonicity' on various manifold. Sometimes, I am counfuesed by the definitions. For real manifold, the harmonic manifold is defined by $\Delta f=0$, where $\Delta$ is the Riemannian Laplacian operator, clearly, we need the metric here to define harmonicity. So the subharmonic function is defined by $\Delta f\le 0$.
Things are somehow changed in the complex case. Where we have $\partial$ and $\bar{\partial}$ operator, which I think can be defined without introducing any Hermitian metric on the complex manifold. Then we defines pluriharmonic or plurisubharmonic to be $i\partial\bar{\partial} f = 0$ or $\ge 0$.
Hence the pluriharmonicity and plurisubharmonicity can be defined without introducing metric?
The key fact is that biholomorphic transformations of a domain in $\mathbb C^n$ preserve pluriharmonic (ph) and plurisubharmonic (psh) functions. That is, $u$ is ph/psh if and only if $u\circ \Phi$ is ph/psh.
Therefore, the ph and psh classes naturally transfer to complex manifolds by means of chart maps. Yes, you can also use the $\partial$ and $\bar \partial$ operators for this purpose. Just keep in mind they produce differential forms, not functions. But there is a concept of positive forms.
The difference with the real case is that in $\mathbb R^n$ diffeomorphisms do not preserve (sub)-harmonicity.
As a matter of fact, biholomorphic maps in $\mathbb C^n$, $n>1$, do not preserve (sub)-harmonicity either, which makes these non-pluri concepts less useful in several complex variables.