The celebrated (1+1)-dimensional Korteweg de–Vries (KdV) equation and its (2+1)-dimensional extension, the Kadomtsev–Petviashvili (KP) equation, are two of the most important models in physical science. The KP hierarchy is explicitly written out by means of the linearized operator of the KP equation. A novel (2+1)-dimensional KdV extension, the cKP3–4 equation, is obtained by combining the third member (KP3, the usual KP equation) and the fourth member (KP4) of the KP hierarchy. The integrability of the cKP3–4 equation is guaranteed by the existence of the Lax pair and dual Lax pair. The cKP3–4 system can be bilinearized by using Hirota’s bilinear operators after introducing an additional auxiliary variable. Exact solutions of the cKP3–4 equation possess some peculiar and interesting properties which are not valid for the KP3 and KP4 equations. For instance, the soliton molecules and the missing D–Alembert type solutions (the arbitrary travelling waves moving in one direction with a fixed model dependent velocity) including periodic kink molecules, periodic kink-antikink molecules, few-cycle solitons, and envelope solitons exist for the cKP3–4 equation but not for the separated KP3 equation and the KP4 equation.