If $X$ is a noncompact LCH space (locally compact, Hausdorff) then its one point compactification is $X^*=X\cup \{\infty\}$ with topology $\mathcal{T^*}$ given by $U \in \mathcal{T^*}$ iff either
a) $U \subset X$ is open, or
b) if $\infty \in U$ then $U^c \subset X$ is compact
What whould be the definition and the topology for 2 point compactification, or 4 point compactification. Like in $\mathbb{R^2}$ where there are 4 infinities: $\pm \infty$ on $x$ axis and $\pm \infty$ on $y$ axis?
On
$R$ has a $1$-point and a $2$-point compactification but not an $n$-point comp... for finite $n>2.$ (This was a problem in AMM some decades ago.) The space $\omega_1$ with the $\epsilon$-order topology has, up to equivalence, only 1 comp... namely, the identity embedding into $\omega_1+1.$ It may be useful in some algebraic contexts to have just 4 infinities for the real plane, on the axes, but topologically the line $y=x$ is going to be confused about which of these it should "go to".
For $n>1$ the space $\Bbb R^n$ has a one-point compactification, but it does not have a $k$-point compactification for any integer $k>1$; see this PDF. $\Bbb R$ itself has both a one-point compactification, which is homeomorphic to the circle $S^1$, and a two-point compactification, which is homeomorphic to $[0,1]$.
You might also want to look at the notion of an end in topology.