I'm reading the section on the extended real number system from Rudin's Principles of Mathematical Analysis, where he introduces the numbers $\infty$ and $-\infty$ and defines how they operate with finite numbers. My intuition is that in general, it's dangerous to expand a set with some operation and then define how the operation acts on the new elements because this opens the door to contradiction with the already established laws from the original set.
For example, if we wanted to expand the Klein four-group $K = \{1,a,b,ab\}$ by introducing an additional element $c$ and defining, say, $ac = ca := b$ and $bc = cb := a$, then we would be able to deduce $abc = a^2 = 1$, and now $ab$ has two inverses. How can we be absolutely certain that nothing equally pathological happens when we introduce $\infty$ and $-\infty$ into the real numbers, or if we expand the complex numbers by adding the single point $\infty$?
There aren't any contradictions introduced, because $\infty$ and $-\infty$ are not added to the real number system. They are introduced as hyperreal numbers $^*\mathbb{R}$ where $$\mathbb{R}\subset{^*\mathbb{R}}$$ and $$\pm{\infty}\in{^*\mathbb{R}}$$ but $$\pm{\infty}\notin{\mathbb{R}}$$ I suppose the word "extended" is misleading. $\mathbb{R}$ is not being modified, rather, a superset of $\mathbb{R}$ is introduced ($^*\mathbb{R}$).