There's certain situations where a message is better off not being seen. Things that are upsetting, rude or offensive for certain, or one might want to housekeep their inbox.
Dopplegreener's post here - gives an example where this would be useful on chat, with a rude message that was deleted turning up on the inbox and here's an example of it being useful on comments, where a seen inbox entry isn't purged. - so in general some kind of mechanism for this would be useful across the public platform components. There's a suggestion we can request the latter as its own question, but this seems a inbox level, not chat/qa level issue.
Implementation obviously should be up to y'all, but a minimum viable option would be something global to a post or chat message, and the ideal would be per user being able to set what they don't want to see.
I'm not sure if the inbox is generated dynamically or managed seperately but being able to mark content as something not to turn up on the inbox might be a useful feature.