I am interested in topological spaces such that whenever the space embeds into the Hilbert cube, 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 "Characterization of finite-dimensional 𝑍-sets" by Kroonenberg [Proc. Amer. Math. Soc. 43 (1974), 421-427].

Are there other examples?