Reading the wikipedia pagina, a complex measure $\mu$ is a function $\mathcal{F} \to \mathbb{C}$ that is $\sigma$-additive. I.e., if $(E_n)_n$ is a set of disjoint sets in the $\sigma$-algebra $\mathcal{F}$, we have
$$\mu(\bigcup_{n=1}^\infty E_n) = \sum_{n=1}^\infty \mu(E_n)$$
Unlike positive measures, we don't require that $\mu(\emptyset) = 0$. Why don't we require this last property?
On
Note that a complex measure is a bounded function, but a positive measure need not be bounded. So whenever $\sigma$ is a positive measure $\sigma(\phi \cup \phi)=\sigma(\phi)+\sigma(\phi)$ doesn't always imply $\sigma(\phi)=0$, due to $\infty+\infty=\infty$.
But for a complex measure $\mu$ we have $\mu(\phi)\in \Bbb C$,hence $\mu(\phi \cup \phi)=\mu(\phi)+\mu(\phi)\implies \mu(\phi)=0$
From $\sigma$-additivity, we have $\mu(A\cup B)=\mu(A)+\mu(B)$ whenever $A\cap B=\emptyset$. Take $A=B=\emptyset$.