  The conformal bridge transformation (CBT) is reviewed in the light of the
$\mathcal{PT}$ symmetry. Originally, the CBT was presented as a non-unitary
transformation (a complex canonical transformation in the classical case) that
relates two different forms of dynamics in the sense of Dirac. Namely, it maps
the asymptotically free form into the harmonically confined form of dynamics
associated with the $\mathfrak{so}(2,1)\cong \mathfrak{sl}(2,{\mathbb R})$
conformal symmetry. However, as the transformation relates the non-Hermitian
operator $i\hat{D}$, where $\hat{D}$ is the generator of dilations, with the
compact Hermitian generator $\hat{\mathcal{J}}_0$ of the
$\mathfrak{sl}(2,{\mathbb R})$ algebra, the CBT generator can be associated
with a $\mathcal{PT}$-symmetric metric. In this work we review the applications
of this transformation for one- and two-dimensional systems, as well as for
systems on a cosmic string background, and for a conformally extended charged
particle in the field of Dirac monopole. We also compare and unify the CBT with
the Darboux transformation. The latter is used to construct
$\mathcal{PT}$-symmetric solutions of the equations of the KdV hierarchy with
the properties of extreme waves. As a new result, by using a modified CBT we
relate the one-dimensional $\mathcal{PT}$-regularized asymptotically free
conformal mechanics model with the $\mathcal{PT}$-regularized version of the de
Alfaro, Fubini and Furlan system.
