When I read Griffths and Harris, the Hodge star operator is an operator form $A^{p,q}(M)$ to $A^{n-p,n-q}(M)$. But in Huybrechts' book it is from $A^{p,q}(M)$ to $A^{n-q,n-p}(M)$, because it takes conjugate before wedge product.
So which definition is more standard? Why does there exist two different definitions? I am really confused by this, thanks!
Some define the Hodge star operator by $\langle \alpha, \beta\rangle\operatorname{vol} = \alpha\wedge\ast\beta$, in which case $\ast : A^{p,q}(M) \to A^{n-p,n-q}(M)$.
Other people instead define the Hodge star operator by $\langle \alpha, \beta\rangle\operatorname{vol} = \alpha\wedge\overline{\ast\beta}$ or $\langle\alpha, \beta\rangle \operatorname{vol} = \alpha\wedge\ast\bar{\beta}$, in which case $\ast : A^{p,q}(M) \to A^{n-q,n-p}(M)$.
The fact that there are different conventions is not deep, it is just a matter of preference.
In my experience, $\langle\alpha, \beta\rangle\operatorname{vol} = \alpha\wedge\overline{\ast\beta}$ is more common.