Almost contact complex Riemannian manifolds, also known as almost contact B-metric manifolds, are equipped with Ricci–Bourguignon-like almost solitons. These almost solitons are a generalization of the known Ricci–Bourguignon almost solitons, in which, in addition to the main metric, the associated metric of the manifold is also involved. In the present paper, the soliton potential is specialized to be pointwise collinear with the Reeb vector field of the manifold structure, as well as torse-forming with respect to the two Levi-Civita connections of the pair of B-metrics. The forms of the Ricci tensor and the scalar curvatures generated by the pair of B-metrics on the studied manifolds with the additional structures have been found. In the three-dimensional case, an explicit example is constructed and some of the properties obtained in the theoretical part are illustrated.