Implement dedicated search feature

I was surprised to be redirected to Google when using the search feature.
Maybe I misremember it, but before the latest redesign the search feature was a self-contained one for the website and docs.

If it is planned to implement a dedicated search in the future it would be also great to consider adding support for [OpenSearch](https://developer.mozilla.org/en-US/docs/Web/OpenSearch) to it. That way it can be used as search provider in browsers directly.