  We study a generalized $\phi^4$ model that gives rise to BPS kink/antikink
configurations with compacton-like profiles. One observes that the positive
parameter controlling the generalizing function promotes an infinity
degenerescence of the BPS solutions. We then use the Differential
Configurational Complexity technique to distinguish the degenerate
configurations, which allows us to obtain the parameter values providing the
most likely field profiles. Besides, the analysis of the excitation spectrum of
the model shows the existence of translational and vibrational modes. Thus, the
emergence of bound states of solitons (bions) and resonance phenomena is
guaranteed when analyzing the scattering of kink/antikink structures. In this
way, one notes that depending on the initial velocity, the collision can be
inelastic or quasi-elastic, even in the case of compacton-like configurations.
