What does infinity in complex analysis even mean?

$\infty$ is the compactification point of the complex plane. In this context, $\infty$ is very real: it is the north pole of the Riemann sphere. $0$ is the south pole.

In fact, $0$ and $\infty$ are closely related:

• they both only have a modulus and no direction or argument.

• the function $z \mapsto \dfrac 1z$ exchanges them.

• they $0$ and $\infty$ are absorbing elements for multiplication: $0 \cdot z = 0$, $\infty \cdot z = \infty$, for $z \ne 0, \infty$.

In the complex plane, it is useful to first think of how we change the meaning of the term "bounded". In the reals, "bounded by $M$" means "$|x| < M$". In the complexes, "bounded by $M$" looks like it means the same thing, "$|z| < M$", but $|\cdot|$ has changed from real magnitude to complex magnitude.

We might say that a sequence in the reals is unbounded if for any bound $M$, the sequence is eventually permanently made up of terms that are greater in magnitude than $M$. (We might, but we shouldn't. We should say "unbounded" means that for any $M$, the absolute values of the sequence at least once exceed $M$, so that no bound is large enough to dominate the entire sequence. However, it is useful to think about sequences that are monotonic to fix one's ideas before thinking about general sequences.) Note that we haven't said if they're increasing towards $+\infty$ or decreasing towards $-\infty$. We've said they're unbounded in magnitude without picking a direction, so the order of the reals isn't entering into this idea. We can do the same thing in the complexes -- if a sequence eventually permanently exceeds any given bound, then the sequence is unbounded in magnitude. If we were going to be technically sound, we would say of sequences of points, $x_1, x_2, \dots$ and $z_1, z_2, \dots$, "$|x_n|$ goes to infinity" or "$|z_n|$ goes to infinity." Again, we are ignoring the order and only attending to the magnitude of the numbers. If this is where we stop, there is only one infinity. Under stereographic projection of the plane onto the unit sphere, that infinity appears as the north pole. See "Riemann sphere", where the circles of larger and larger magnitude are eventually drawn up (like a bag) to a single point, labelled "$\infty$". When we treat infinity in the complexes this way, it is easier to talk about a "neighborhood of infinity" -- it's a disk of points around the north pole in the projection. When we un-project, that's the set of points outside a disk centered at $0$. One reason this way of thinking of infinity is useful is that now every point on the sphere "is the same" -- any little neighborhood you pick around one point can be slid along the sphere to another point, even the point at infinity. So computations that previously only made sense at finite points can be made sensical at $\infty$.

If we do pay attention to the order, we can be a little more precise and, in the reals, say "$x_n$ goes to $+\infty$" if for any bound $M>0$, the sequence eventually permanently above that bound and say "$x_n$ goes to $-\infty$" if for any bound $M<0$, the sequence is eventually permanently below that bound. If we pretend that these sequences are sequences on the real line in the complexes, then we see that in the one case, we are going in the direction $\mathrm{e}^{\mathrm{i} 0} = 1$ and in the direction $\mathrm{e}^{\mathrm{i} \pi} = -1$ and notice that we said we were going to $1 \cdot \infty$ and $-1 \cdot \infty$, if we allow ourselves (only) the tiniest amount of algebra with infinities. But we can take a sequence $x_n$ going to $+\infty$ and make a new sequence $z_n = \mathrm{e}^{\mathrm{i} \theta} x_n$, which is a sequence whose magnitude increases without bound and whose direction is along the ray starting at $0$ and passing through $\mathrm{e}^{\mathrm{i} \theta}$. Could we not generalize our idea of infinity to allow directed infinities? "$z_n$ goes to $\mathrm{e}^{\mathrm{i} \theta} \infty$". This idea expresses that for any bound $M>0$ and any tiny error in the argument $\delta \theta>0$, the sequence eventually permanently has $|z_n| > M$ and $\arg(z_n) \in (\theta - \delta \theta, \theta + \delta \theta)$. Now we can more precisely describe sequences that run away from the origin asymptotically approaching a ray from the origin -- a form of going to a signed infinity where the sign can be any point on the complex unit circle, just as it was for signed infinities in the reals (where there were only two points on the unit circle, $\pm 1$).

We can generalize further, and this process of deciding what to do to sequences that try to "run off the edge" (whatever that means for infinite spaces having no concrete edges) leads to points at infinity. There are many choices. The Alexandroff one-point compactification makes the choice that there is only one $\infty$, like our magnitude choice above, and chooses to treat all eventually permanently exceeding any given bound sequences as all converging to the "one point", $\infty$. This gives variations of the Riemann sphere for whatever space you do it to. If we follow the directed infinity path, then we get various projective spaces, but at least we can keep track of the direction in which we ran away to infinity. There are additional ways to collect sequences running away to infinity into classes and assign a "point at infinity" for the members of that class, leading to various compactification schemes, but this is probably more generalization than you need for your current thoughts on infinity.

There is no order in $\mathbb C$. Right! But if $z\in\mathbb C$, $|z|$ can be big or it can be close to $0$. So, take a sequence $(z_n)_{n\in\mathbb N}$ of complex numbers. Suppose that $\lim_{n\to\infty}|z_n|=+\infty$. What this means is that the (real) numbers $|z_n|$ are arbitrarily large. So, you can state this saying that $\lim_{n\to\infty}z_n=\infty$ (not $+\infty$).

Another example. Let $f(z)=\frac 1z$. If $z$ is very far from $0$ (that is, if $|z|$ is very large), then $f(z)$ is very close to $0$. We express this saying that $\lim_{z\to\infty}f(z)=0$. The formal definition is$$(\forall\varepsilon>0)(\exists R>0):|z|>R\implies\bigl|f(z)\bigr|<\varepsilon.$$

Note that in $\mathbb C$ we only have one $\infty$; we don't have a $+\infty$ and a $-\infty$.

For more about this, read about the Riemann sphere.