The set $A= [z \in \mathbb C : |z+2|<2$ or $|z-2|<2]$ is not path connected.
I supposed that $A$ is path connected. Then there is a path , a continuous function $\gamma (t)=f_1(t)+if_2(t)$.
After that assumption how could I arrive at a contradiction ?
First, notice that $B(-2,2)$ and $B(2,2)$ are convex and so path connected, so to show that $A$ is not path connected, you must show that there is no path between the two sets.
Suppose $\gamma$ is such a path, that is $\gamma(0) \in B(-2,2), \gamma(1) \in B(2,2)$.
Look at $r(t)=\operatorname{re} \gamma(t)$. Note that $r(0) <0, r(1) >1$ so there must be some $t$ such that $r(t) = 0$. However, the inaginary axis does not intersect $A$ hence a contradiction.
Intersecting the balls of radius slightly greater than the balls you have with your subspace implies that your subspace isn't connected (since you will have two disjoint open non-trivial sets which cover the space).
Now, not connected implies not path-connected.
