Dominated convergence theorem for complex-valued functions?

Use bounded convergence theorem for each coordinate. Since $|f_n|\leq g$ implies $|\Re f_n| \leq g$ and $|\Im f_n| \leq g$ ($\Re z$ is the real part of $z $ for $z \in \Bbb{C}$, and $\Im z$ is the immaginary part of $z$) $$\Re\bigg(\int f_n (x)\, dx\bigg) = \int \Re f_n(x)\, dx \\ \Im\bigg(\int f_n (x)\, dx\bigg) = \int \Im f_n(x)\, dx$$

Then the bounded convergence theorem yields $$\lim_n\Re\bigg(\int f_n (x)\, dx\bigg) = \int \lim_n\Re f_n(x)\, dx \\ \lim_n\Im\bigg(\int f_n (x)\, dx\bigg) = \int \lim_n\Im f_n(x)\, dx$$

And conclude noting that

$$\lim_n\int f_n (x)\, dx = \lim_n\bigg(\Re\bigg(\int f_n (x)\, dx\bigg) + i \Im\bigg(\int f_n (x)\, dx\bigg)\bigg) \\= \int \lim_n\Re f_n (x)\, dx +i \int \lim_n\Im f_n(x)\, dx $$