Add to ORKGQuantum cosmology and the quantum wormhole with a conformal complex scalar field are discussed, the corresponding Wheeler-DeWitt equations are obtained, and the cosmological wave functions and wormhole wave functions are calculated, respectively, with different boundary conditions. From the cosmological wave function it is found that the probability density of the universe is zero at a = 0, while at the ground state the most probable radius is about the Planck scale. It is also shown that there exist two different types of universes, which can be connected by the quantum tunneling effect, transiting from one region to another. It follows from the wormhole wave function that the most probable radius of the wormholes is about the Planck scale...