It is hard to understand the formulation of your post, 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

Edit: To those with editing power: It took me a longer while to realize that OP is (probably) asking what happens when in the definition of a hull the class of holomorphic functions is replaced by the class of $\mathbb{C}$-linear function, not quite the question I have just answered. It would benefit from editing.