It'll make navigation way easier.
If we want to do it purely client-side we need to generate a search index and add some JS (ick) to use it. Investigate how Python's docs (or Sphinx in general) does it.
It is imperfect but a reasonable start.
For sphinx's stuff, see:
You can edit DDG's styling a little, see https://duckduckgo.com/params
The more I think about it, the more I like DDG, but we might be able to make the style fit a little better. The drop-shadows in the search box are startling compared to the rest of the site. So, let's make our own form thingy that submits to their page (if that's allowed?) and we can call it a day.
Done in 142:59c483bbdb9d
Though, JS. Meh. Would be happier if we did it purely through forms, but that doesn't seem feasible. Oh well.