Integrating Elasticsearch as a Scout driver seems not very difficult. Just pull in the driver from ErickTamayo:
composer require tamayo/laravel-scout-elastic
After that, add the correspsonding entries to the config/app.php and config/scout.php file, like mentioned on the package's homepage. From now on you could use the Laravel Scout API as @JeffreyWay does in the tutorial videos.
Nevertheless, getting something to work like the Algolia instant search stuff running with Elasticsearch is more sophisticated. So far I couldn't find any package which provides as a similar functionality as the vue-instantsearch (I also think its name should reflect it's using Algolia only).
I was able to create a Vue component that is triggering my search backend on my Laravel app on every @keyup event, but I'm missing the cool stuff like refinement lists and proper highlighting of the query within the results (when searching for more than one word at once).
Here I would appreciate a tutorial.