Algolia has come out with a new WordPress plugin that improves upon built-in search. It's a hosted service (unlike SearchWP and other plugins), so that may deter or attract you depending on your needs.
The plugin promises “out of the box” autocomplete and drop-down menus which look impressive from the screenshots. It looks pretty extensible, and they are planning to release a WooCommerce extension in the near future.
When researching this, I found out WooCommerce.com uses Algolia for delivering search results — though they don't yet use this plugin. I had not heard of Algolia at all until now, but a bunch of big brands are using it. I'm glad to see they've put so much effort into an official WordPress plugin.