The official documentation of TYPO3's Indexed Search mentions that the Crawler extension is mandatory for Indexed Search. Unfortunately Crawler is still not available for TYPO3 v13.4.
Is there an alternative extension or solution to guarantee the functionality of TYPO3's core extension "Indexed Search"?
EXT:crawler 12.0.6 and newer, released on January 24, 2025, is compatible with TYPO3 13.4: https://github.com/tomasnorre/crawler/releases/tag/12.0.6
If you're looking for an alternative solution, have a look at EXT:ke_search: https://extensions.typo3.org/extension/ke_search/
I'm not sure if ke_search supports all the features that indexed_search has, but instead it features things like faceted search. And it't doesn't need a crawler extension.