I think he is referring to the Coulomb Gas formalism.
The 'usual' massless boson in 2D has the energy momentum tensor $T= -\frac{1}{\alpha}\partial X^\mu\partial X_\mu$ (normal ordered). This energy-momentum tensor is the generator of conformal transformations. The current $\partial X^\mu$ and vertex operator $e^{i\alpha X^\mu}$ are examples of fields which transform covariantly with respect to these conformal transformations. These fields have a certain conformal dimension (the current has $h=1$ and the vertex operators something like $h=\frac{1}{2\alpha^2}$, depending on your conventions) and the theory has a central charge of $c=1$.
However, we can modify the energy momentum tensor by adding a term:
$$T= -\frac{1}{\alpha}\partial X^\mu\partial X_\mu + V_\mu \partial^2 X^\mu$$
This energy-momentum tensor also generates conformal transformations, but a different kind. For instance, the field $\partial X^\mu$ is no longer primary (does not transform covariantly with respect to these transformations). The EM tensor in complex coordinates is: $$ T(z) =-\frac{1}{2} :~\partial X^\mu \partial X^\mu~: +i\sqrt{2}\alpha_\mu\partial^2X^\mu$$
where $\alpha_\mu$ is related to $V_\mu$. 298
This extra term arises due to the addition of a linear, imaginary coupling to the theory.
In this new conformal theory the vertex operators are still primary, but with a shifted conformal dimension: $h=\alpha^2 - 2\alpha \alpha_0$. Another thing that can be checked is the central charge, which has been shifted to: $c=1-24\alpha_0^2$. A last interesting thing is that this new theory has a different neutrality condition.
Moral of the story: by the addition of an extra, linear coupling to theory we move away from the original massless boson. However, we do not lose the conformal symmetry, but instead uncover a whole zoo of conformal theories. The massless boson does not just realize the $c=1$ case, but can be used to construct many, many different central charges and different content of primary fields. For example, with this construction you can obtain all the minimal models.
See chapter 9 of the big yellow book by <PERSON>, <PERSON> and <PERSON>. 976