We study the interaction of chimera states in multiplex two-layer systems, where each layer represents a network of interacting phase oscillators with adaptive couplings. A feature of this study is the consideration of synchronization processes for a wide range of chimeras with essentially different properties, which are achieved due to the use of different types of coupling adaptation within isolated layers. We study the effect of forced synchronization of chimera states under unidirectional action between layers. This process is accompanied not only by changes in the frequency characteristics of the oscillators, but also by the transformation of the structure of interactions within the slave layer that become close to the properties of the master layer of the system. We show that synchronization close to identical is possible, even in the case of interaction of chimeras with essentially different structural properties (number and size of coherent clusters) formed by means of a relatively large parameter mismatch between the layers. In the case of mutual action of the layers in chimera states, we found a number of new scenarios of the multiplex system behavior along with those already known, when identical or different chimeras appear in both layers. In particular, we have shown that a fairly weak interlayer coupling can lead to suppression of the chimera state when one or both layers of the system demonstrate an incoherent state. On the contrary, a strong interlayer coupling provides a complete synchronization of the layer dynamics, accompanied by the appearance of multicluster states in the system's layers.