Let $K$ be compact in $\Bbb C^n$, without loss of generality $K$ connected.
We define its polynomially convex hull as $$ \widehat{K}:=\{z\in\Bbb C^n\;:\;|f(z)|\le\|f\|_K\;\forall f\in\mathcal O(\Bbb C^n)\} $$
I think $\widehat{K}$ has to be connected as well (and in general, has to have the same number of connected components as $K$), but I don't know nor if it's true, neither where to start from.
If $n=1$ the answer is positive, since the $\widehat K=K\cup\{\mbox{holes of}\;K \}$.
What about $n>1$? I suspect it is something well known, therefore a reference would be perfect.