Let $U\subset{\mathbb{C}}$ be open and $c=n_1•c_1+…+n_k•c_n$ (formal sum with $c_i:[0,1] \to U$ continuous, $n_i \in \mathbb{Z}$) be a null-homologous cycle (closed 1-chain with $Ind_c(z)=0 \text{ for } z \in \mathbb{C}\setminus U$) in $U$. Since the support $|c|=\cup_i c_i([0,1])$ is compact there exist a $R>0$ such that the support is contained in an the open ball around $0$ with radius $R$, i.e. $|c| \subset B_R(0)$.
Question: Is then the winding number zero for all elements outside this ball,I.e. $Ind_c(z)=0$ for $z \in \mathbb{C} \setminus B_R(0)$?
( I know that the winding number is constant on connected components in the complement of the support, so since the complement of the ball is convex the winding number is constant there and if there exists a $z \in \mathbb{C}\setminus (U \cup B_R(0))$ the winding number is zero outside the ball. But what if $(\mathbb{C}\setminus B_R(0)) \subset U$?)
As you said, the winding number is constant on each component of the complement of the support of the cycle, so it is constant on $\mathbb{C} \setminus B_R(0)$. Also $\lim_{z \to \infty} \operatorname{Ind}_c(z) = 0$, and therefore that constant is zero.
In other words: $\operatorname{Ind}_c(z)$ is zero for all $z$ in the unbounded component of the complement of the support of the cycle.