According to Peter Cameron (in his "Notes On Classical Groups", available on his webpage) "Paul Cohn constructed an example of a division ring $F$ such that all elements of $F\backslash\{0,1\}$ are conjugate in the multiplicative group of F." This in particular implies that the group can be chosen to be the multiplicative group of a division ring.

According to Peter Cameron (in his "Notes On Classical Groups", available on his webpage) "Paul Cohn constructed an example of a division ring $F$ such that all elements of $F\backslash\{0,1\}$ are conjugate in the multiplicative group of F." This in particular implies that the group can be chosen to be the multiplicative group of a division ring.

added:

I again thank Yves Cornulier for providing the reference to Cohn's paper. The credit shoud be given to him.\

An easy but interesting fact: Let $G$ be a group and consider the {\it diagonal action} of $G\times G$ on the set $X=G$ by

$$(g,h)\cdot x=g^{-1}xh.$$ It can be shown that this action is $2$-transitive if and only if all non-identity elements of $G$ are conjugate. (This is an exercise in {\it P. J. Cameron, Permutation groups, LMS, 1999.})