Yes, any nonempty open domain $ D\subset \mathbb C$ is a domain of holomorphy.
The proof is in two steps:
a) One proves that a holomorphically convex domain $D\subset \mathbb C^n$ is a domain of holomorphy .
This is not difficult: see for example Grauert-Fritzsche, Theorem 6.5, page 81
b) One proves that for $n=1$ any domain $D\subset \mathbb C$ is holomorphically convex.
This too is rather easy: given a compact subset $K\subset\subset D$ and a boundary point $a\in \partial D$, consideration of $\frac{1}{z-a}$ shows that the holomorphically convex hull $\hat {K}=\hat {K}_D$ cannot approach $\partial D$ and consideration of the holomorphic function $z$ shows that $\hat {K}$ is bounded.
Since $\hat {K}$ is closed in $D$ these considerations prove that $\hat {K}$ is compact.
NB Actually a) above is an equivalence: a domain $D\subset \mathbb C^n$ is holomorphically convex iff it is a domain of holomorphy.
This result was proved by Cartan-Thullen in 1932. Here are the authors reunited 55 years later.