It is shown that the representation space of the conformal groupSO4,2 is the Bargmann-Segal space of coherent states |wμ› of the non-Hermitian position operator\(\bar w_\mu ^ + \). The states |wμ› are coherent in the usual sense of minimal uncertainty ofxμ andpμ. The real part of\(\bar w_\mu ^ + \) gives the Hermitian position operatorxμ. Both operators fulfil the canonical commutation relations and are covariant under the conformal group. The imaginary part of\(\bar w_\mu ^ + \) describes the dispersion of position. The metric operatorG connecting\(\bar w_\mu \) with its adjoint\(\bar w_\mu ^ + \) gives the scalar product for various representations of the conformal group.