In the book "A Mathematical Introduction to Conformal Field Theory" by M. Schottenloher, the author says in in Section 2.1:
To study the collection of all conformal transformations on an open connected subset $M\subset \mathbb{R}^{p,q}$, $p+q\geq 2$, a conformal compactification $N^{p,q}$ of $\mathbb{R}^{p,q}$ is introduced, in such a way that the conformal transformations $M\to \mathbb{R}^{p,q}$ become everywhere defined and bijective maps $N^{p,q}\to N^{p,q}$. Consequently, we search for a "minimal" compactification $N^{p,q}$ of $\mathbb{R}^{p,q}$ with a natural semi-Riemannian metric, such that every conformal transformation $\varphi : M\to \mathbb{R}^{p,q}$ has a continuation to $N^{p,q}$ as a conformal diffeomorphism $\hat{\varphi}:N^{p,q}\to N^{p,q}$.
Now I would like to understand intuitively why is this necessary. What is the motivation to pass to a conformal compactification?
I mean, can't one simply focus on conformal diffeomorphisms $\varphi : \mathbb{R}^{p,q}\to \mathbb{R}^{p,q}$? What is the obstruction for doing this?
Why introduce a conformal compactification here? What is the motivation and how can we understand such need intuitively?