= Solution
In the compact <twistor correspondence>, a point $p$ of <complexified compactified Minkowski space> is a two-dimensional subspace $W_p\subset\mathbb T$. The correspondence fiber consists of flags $(\ell,W_p)$ with $\ell$ a one-dimensional subspace of $W_p$. Its image in <projective twistor space> is precisely $\mathbb P(W_p)$, so
$$
\boxed{\widehat p=\mathbb P(W_p)\cong\mathbb{CP}^1.}
$$
This is the <twistor line> of $p$. In the affine chart the <twistor incidence relation> gives $W_p=\{(ip^{AA'}\pi_{A'},\pi_{A'}):\pi\in\mathbb C^2\}$. The linear map $\pi\mapsto(ip\pi,\pi)$ is injective, and projectivizing it gives the same <Complex projective space> $\mathbb{CP}^1$. The subspace argument also covers points at conformal infinity, where that graph chart fails.
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