At the moment Adobe Live Search for Magento only supports searching for products. I'd like to suggest it be extended to support searching for CMS page content as well. Our website has lots of content related to our products and it would be great to have that content show up within the search results for customers to find. Our current search provider Algolia can serve up CMS content results along with products in the same search query. In the search pop up, I'd like to see CMS search results in the left column and products shown in the right.
In addition, it would be great if Live search could be extended to support indexing of 3rd party extensions for additional searchable content. For example, my company sells books. We created a custom module that databases authors who wrote the books we sell. This database is displayed to customers on the front-end of the site. Each author has page with their biography and a list of active products. Authors are also linked to from the product page. I'd love to be able to create a custom Live Search extension the would allow our authors to be searched on by name.
Another example is Blog software. We use a blogging extension provided by a premium Magento partner and having their blog content showing up within site search would be great as well.