Compared to metamaterials based on regulation of real material parameters, artificial materials designed by controlling complex parameters exhibit richer physics and different functions. In this work, we present acoustic conjugate metamaterials, which possess complex mass densities and bulk compressibilities with opposite phases, to manipulate acoustic waves. By adjusting the phases of the complex material parameters, we can change the handedness and loss/gain of the material, and then, produce unique phenomena in acoustic fields. Theoretical analysis and numerical simulations demonstrate that placing a slab of an acoustic conjugate metamaterial in a traditional medium can create acoustic laser modes, perfect absorption or coherent perfect absorption. Moreover, even if total reflection arises at the interface, extraordinary transmission is observed, which differs from the phenomenon of traditional total reflection. Therefore, acoustic conjugate metamaterials exhibit unprecedented functions that have not been observed in the metamaterials on account of real parameter manipulation.