This would not suit all users and so I hope any implementation would be opt-in and reversible.
All mastodon accounts have a built-in RSS feed (add .rss to end of url). If those feeds were aggregated to a public database that could be searched, there is a searchable record of mastodon. Or, a private database of only the accounts one wants to follow and search.
I have a private rss-based record of my own posts at mastodon that I can search through by way of good old IFTTT.com.
Church of the SubGenius Members-Only MastoDobbs.