There is a related iterated function system with two functions,
$f_0(x) = 1+zx$
$f_1(x) = -1+zx$
$X_z$ is the unique nonempty compact fixed set of this iterated function system. It is sometimes called a generalized dragon set with parameter $z$, and particular values of $z$ can produce some well-known fractals called dragons.
A relevant result on iterated function systems is that the fixed set $X$ is connected iff it is arcwise connected iff the family of subsets $\{f_i(X)\}$ is connected, which in this case means $f_0(X_z) \cap f_1(X_z)$ is nonempty. (This paper refers to Kigami, Analysis on Fractals chapter 1 for the result.) So, the set is connected iff we can write
$$\begin{eqnarray} 1 + \sum_{i=1}^{\infty} a_i z^i &=& -1 + \sum_{i=1}^\infty a_i' z^i \newline 1 + \sum_{i=1}^{\infty} \frac{a_i-a_i'}{2} z^i &=& 0 \newline 1 + \sum_{i=1}^{\infty} b_i z^i &=& 0 \end{eqnarray}$$
where $b_i = (a_i-a_i')/2 \in \{0,1,-1\}$.
In particular, $X_z$ is connected when $z$ is real with $1/2 \le |z| \lt 1$ and when $z$ is a root of a polynomial with coefficients in $\{-1,0,1\}$. The intersection of the closure of those roots with the interior of the disk is the entire set where $X_z$ is connected.
The closures of roots of polynomials with restricted coefficients have been studied, and they are quite interesting. Often, the boundary of the closure locally resembles the fixed set of the iterated function system of that point.