A special coupled KdV equation is proved to be the Painlevé property by the Kruskal's simplification of WTC method. In order to search new exact solutions of the coupled KdV equation, Hirota's bilinear direct method and the conjugate complex number method of exponential functions are applied to this system. As a result, new analytical complexiton and soliton solutions are obtained synchronously in a physical field. Then their structures, time evolution and interaction properties are further discussed graphically.