Why is a complex number plus infinity equal to infinity?

Question

Why is $$2 + 3 i + \infty = \infty$$ according to Mathematica and Wolfram Alpha?

Shouldn't it be:

$$2 + 3 i + \infty = \infty + 3 i$$

?

After all:

$$2 + 3 i + 10 = 12 + 3 i$$

and not:

$$2 + 3 i + 10 = 12$$

Answer 1

In the context of the complex plane there is a very useful notion of a "point at infinity" but it is in infinity "in every direction". This gives rise to a notion of the Riemann Sphere, where two copies of the complex plane give coordinate charts with transition $z\mapsto 1/z$. Complex functions with poles can then be viewed as functions from the Riemann Sphere to itself satisfying certain niceness conditions.