Im 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 quasicompact groups what some other people refer to simply as compact groups unfortunately  In the latter case the homomorphism iota will be injective thus iotaG 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 iotaG as a topological group but Im sure theres 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 Ive not actually stumbled across an actual construction except in the abelian case