When is a CPD weighted shift similar to a subnormal operator?

We prove that a CPD unilateral weighted shift $W_{\lambda}$ of type III is a quasi-affine transform of the operator $M_z$ of multiplication by the independent variable on the $L^2(\rho)$-closure of analytic complex polynomials on the complex plane, where $\rho$ is a measure precisely determined by $W_{\lambda}$. By using this model, we provide necessary and sufficient conditions for similarity of $W_{\lambda}$ to $M_z$. Necessary conditions for a CPD operator to be similar to a subnormal one are given. A variety of concrete classes of non-subnormal CPD unilateral weighted shifts similar to subnormal operators are established.