It is hard to understand the formulation of your post, and it seems you got some implications wrong, but let me attempt an answer anyway.
I will use the notion of a domain of holomorphy: a domain $\Omega \subset \mathbb{C}^n$, $n \geq 1$, is a domain of holomorphy if there exists a function $f$ holomorphic in $\Omega$ which does not extend holomorphically to any larger domain. Cartan-Thullen theorem says that for a domain in $\mathbb{C}^n$, $n \geq 1$, being a domain of holomorphy is equivalent to being holomorphically convex. It can be proved (using Hahn-Banach theorem) that every convex domain is a domain of holomorphy, so it is holomorphically convex. This is really interesting when $n \geq 2$, because when $n=1$ any domain is a domain of holomorphy. A visualization of a strange-looking domain of holomorphy with $n >1$, aptly called a worm, first described in the paper Diederich, Klas; Fornaess, John Erik A strange bounded smooth domain of holomorphy. Bull. Amer. Math. Soc. 82 (1976), no. 1, 74–76, can be seen here (in a note by Harold Boas)
http://www.ams.org/notices/200305/what-is.pdf
Also, when $n \geq 2$, polynomial convexity is no longer equivalent to simple connectedness. Counterexamples (going both ways) can be found in the book MR1818167 Nishino, Toshio Function theory in several complex variables. (English summary) Translated from the 1996 Japanese original by Norman Levenberg and Hiroshi Yamaguchi. Translations of Mathematical Monographs, 193. American Mathematical Society, Providence, RI, 2001. xiv+366 pp. ISBN: 0-8218-0816-8