||In the iron-arsenide pnictide compounds CaFe2As2 and BaFe2As2, superconductivity can be induced either by a variation of its chemical composition, e.g., by replacing a small percentage of Fe ions by Co ions, or by a reduction of the unit-cell volume through the application of hydrostatic pressure. Both methods have a direct impact on the charge carrier concentration and can be regarded as a way of doping the samples. To understand the nature of superconductivity in this so-called ”122” pnictide system better, we performed specific-heat and electrical-transport measurements under the influence of high pressure on three different samples: CaFe2As2 and Ba(Fe1-xCox)2As2, where x = 5.5% and 3.75%. The characteristic specific-heat jumps at the superconducting transitions, which have been observed in all of these three samples, are a clear proof of the existence of pressure-induced bulk superconductivity. However, some indications are found that the superconducting state in underdoped samples may be inhomogeneous. Furthermore, several additional phase transitions are observed in the specific heat in the overdoped region of the phase diagram, which are most likely related to a structural and a magnetic transition which approach the superconducting transition line close to optimal doping. The collected data suggests that the order parameters of these magnetic, structural and superconducting transitions are strongly ’entangled’ in the superconducting phase and this entanglement needs to be considered in evolving theories on the mechanism of superconductivity in the iron pnictide superconductors.