`  I'm a bit confused about this.  As I understand it, the Bohr compactification can be defined using either only Hausdorff compact groups (what some people simply refer to simply as compact groups) or quasi-compact groups (what some other people refer to simply as compact groups, unfortunately).  In the latter case, the homomorphism iota will be injective, thus iota(G) is a dense subgroup of H which is algebraically isomorphic to G, but carries a coarser topology in general.  I am not aware of a simple characterisation of those groups for which G is isomorphic to iota(G) as a topological group, but I'm sure there's a list of sufficient conditions around.  I suppose that the usual construction of the Bohr compactification is something along the lines of ``the closure of the diagonal in the direct product of all compact groups in which G maps via a continuous homomorphism with dense image``?  That might be easy enough to include in the article, but so far I've not actually stumbled across an actual construction, except in the abelian case.   `