Let $G$ be an infinite group such that for every $g\neq 1$ in $G$, there is some $b$ such that $C(b)$ has finite index in $G$ and $g$ is not in $C(b)$. Is something known about those groups? Are they FC groups, or related to FC groups?



Groups with trivial centralizer-connected component
Let $I$ be an infinite index set and for $i \in I$ let $G\_i$ be a finite group with trivial centre. Let $G:=\prod\_{i \in I} G\_i$ be the direct product of these groups.
Then $G$ satisfies the property you mentioned
$$ \forall g \in G\backslash \lbrace 1 \rbrace \exists b \in G: [G:C(b)]<\infty, g \not\in C(b) $$
 but is not an FC-group, i.e., not every conjugacy class is finite. 


Indeed, let $g=(g\_i)\_{i \in I}\neq 1$ and $i\_0 \in I$ such that $g\_{i\_0}\neq 1$. Let $b=(b\_i)\_{i \in I}$ be such that $b\_i=1$ for $i \neq i\_0$ and $b\_{i\_0}$ be such that it does not commute with $g\_{i\_0}$ (which exists since $G\_{i\_0}$ is assumed to have trvial centre). 
Then $g\not\in C(b)$, but $C(b)$ contains $\prod\_{i \neq i\_0} G\_i \times \lbrace 1 \rbrace$
which is of index $|G\_{i\_0}|$.


On the other hand, every element $g=(g\_i)\_{i \in I} \in G$ with infinite support, i.e. $g\_i\neq 1$ for infinitely many $i$, clearly has infinitely many conjugates. 


