  Remarkably, the twistor $\mathbf P^1$ occurs as a fundamental object in both
four-dimensional space-time geometry and in number theory. In Euclidean
signature twistor theory it is how one describes space-time points. In recent
work by Fargues and Scholze on the local Langlands conjecture using geometric
Langlands on the Fargues-Fontaine curve, the twistor $\mathbf P^1$ appears as
the analog of this curve at the infinite prime.
  These notes are purely expository, written with the goal of explaining, in a
form accessible to both mathematicians and physicists, various different ways
in which the twistor $\mathbf P^1$ makes an appearance, often as a geometric
avatar of the quaternions.
