I am interested in topological spaces such that whenever the space embeds into the <a href="http://en.wikipedia.org/wiki/Hilbert_cube">Hilbert cube</a>, the image of the embedding has a path-connected complement.

Any finite dimensional space has this property by an argument based on Alexander duality in a finite dimensional approximation of the Hilbert cube, see e.g. Lemma 2.1 in <a href="http://www.ams.org/journals/proc/1974-043-02/S0002-9939-1974-0334221-8/">"Characterization of finite-dimensional 𝑍-sets"</a>
by Kroonenberg [Proc. Amer. Math. Soc. 43 (1974), 421-427].

Are there other examples?