Probably as an alternative to the list of related videos displayed in a sidebar, YouTube experiments with a search box placed below the video's description. The search box doesn't load a new page to show the results, as you can see in the screenshot.
You can search for things mentioned in the video or create a small playlist of related videos by adding some of the results to YouTube's Quicklist. Maybe in the future YouTube will open the sidebar modules to developers so you can add useful information related to the current video: music lyrics, popular playlists that include a video or videos recorded on the same location.
Update: Another experiment shows a link for a related video and an option to automatically play all the related videos.
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