The autocomplete for the "project" field is appearing in the place where I'd expect autocomplete for the main search bar to appear. This only appears when I use the Timeless skin, I couldn't reproduce it in Vector.
Okay, so they're both class=suggestions, which is fine, except there are also no ids or anything to distinguish between them. They also appear in the DOM in the exact same place, so... we need to differentiate between the different suggestions somehow. Probably by adding an actual id to the skinnav search, as that's the one that needs the special behaviour (placement, fixed position) but that one is generated as part of core, not the skin, so I'm not really sure how to add it.
Not just a MediaWiki-extensions-PageAssessments issue but easily reproducible whereever a special page provides a search field to search (e.g. T218565 used Special:PageMigration, a special page provided by MediaWiki-extensions-Translate, and thus available on MediaWiki.org for example for privileged users). This issue is due to Timeless' CSS rules which, as noted in T218565, are needed to fix a different bug; this problematic behavior is also visible on pages where the mediawiki.userSuggest RL module is used to allow to search for a user account (e.g. SocialProfile's Special:GiveGift).
As mentioned in T218565, one major issue is that the div.suggestions elements are created outside the skin's main container element ( div#mw-wrapper), which makes targeting the CSS...difficult, to say the least. cc'ing local JS wizard @matmarex for thoughts on how to best solve this.