Consider the way $\widehat{\mathbb R}$ relates to $\overline{\mathbb R}$. This set would relate to $\widehat{\mathbb C}$ in a similar way, with $\aleph_1$ infinities, each at different angles, forming somewhat of a circle with an infinite radius encapsulating the complex plane.
Such a set could possibly be defined as: $$ \overline{\mathbb C}=\mathbb C \cup \left\{ x : (\exists \theta \in [0,2\pi)) \left[x=\lim_{r\to\infty} re^{i\theta} \right] \right\} $$
I'm not sure that the above is a rigorous definition, but I feel like it gets the point across. Is there some way this structure could be rigorously defined, and does it have a conventional name?
Note that I am not talking about $\widehat{\mathbb C}$, which contains a single point for infinity akin to the projectively extended real line.
I think the thing you're talking about is very similar $\Bbb RP^2$, the real projective 2-space. There's a point at infinity for each possible "direction" in the plane.
The distinction is that in $\Bbb RP^2$, the point at infinity for lines at angle $\theta$ is the same as that for lines at angle $\theta + \pi$. So arguably, the thing you're really getting at is called "the closed unit disk", with points on $\partial D$ corresponding to your points at infinity. But it's the disk with some underlying geometry, etc., that's not the one from the standard embedding.
This has actually been studied pretty carefully, in a PhD thesis by Jorge Stolfi at Stanford. It's called Oriented Projective Geometry, and I think it was published by IEEE, but it's been a long time, so I'm not certain of that last part. Anyhow, there's a solid reference for you. Here's a link to it on Amazon: https://www.amazon.com/Oriented-Projective-Geometry-Framework-Computations/dp/148324704X