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Let $a in mathbb{C}$. Ahlfors says we let $a + infty = infty$ and $a cdot infty = infty$. But we cannot define $infty + infty$ without violating the laws of arithimetic (i.e. field axioms).

I don't see why this is. Don't we have $infty + infty = infty$ by applying the distributive law to $2cdot infty$? What am I misunderstanding?

Ah, but are you sure you have the distributive law? :)

The proof of non-existence is that the arithmetic operations are defined by continuous extension. So, we need to check whether or not the following limit exists:
$$ lim_{(x,y) mapsto (infty, infty)} x + y $$
If it existed, we could compute it by taking the limit a particular path. The first chooses $x=y$, and the second $x=-y$:
$$ lim_{(x,y) mapsto (infty, infty)} x + y
= lim_{x mapsto infty} x + x = infty $$
$$ lim_{(x,y) mapsto (infty, infty)} x + y
= lim_{x mapsto infty} x - x = 0 $$

The misunderstanding is assuming that the distributive law applies without exception to the extended arithmetic you're defining here. It doesn't, and your reasoning shows why.

Without defining $infty+infty$ you can make the field axioms hold to the extent that the expressions in the axioms are defined at all. The point of not defining $infty+infty$ is exactly to avoid expressions for which different applications of the field axioms would give multiple inconsistent values.

Assume you had such a $infty$ consistant with field axioms. Then $ infty = infty + infty Longrightarrow infty - infty = 0$. This follows from the fact that in a field, a unique additive inverse exists for every element, so that also exists for $ infty$. You wanted to define $infty = infty + infty$, and if we add the additive inverse on both sides, we get $ 0 = infty + (-infty) = (infty + infty) + (-infty)$. By associativity, we can swap the brackets on the right to couple $infty$ with its additive inverse and get $infty + 0 = infty$

But then $ 1 = 1 + infty + (-infty) = infty + (-infty) = 0$, a contradicition.

It has to do with Dimension I think. For example, we can let cos(75 degrees) be cos 45 degrees plus cos 30 degrees, because functions act different way than whole numbers, we can not factor them out. Same thing with Infinity, does it act the same way with functions and whole numbers ? The answer is no. And are the infinity are all the same. What if one infinity is in area, one in linear and another one in cube. Therefore, infinity plus infinity square equals to infinity cube ? It doesn't work that way. If we can solve this problem out, I think we can create a wormhole to go from one side to another side in a shortcut. And infinity has to do with time too, does infinity has the same time mass, we do not know ?