What version are you running?
4.0.4
What's the URL of the page this enhancement relates to, if any?
Any page that has the search box in the top right
Describe the enhancement and the motivation for it.
As you'll know, users can use the search box for either Quick Search or Full-Text Search. These each query a pre-defined set of fields on review requests. Currently, there's no way for an extension to include extra fields, such as custom fields.
A use case is when making an integration with some other system, and a custom field stores identifiers of related entries in the other system. Then users would be able to search for reviews using those identifiers.
One way to do this would be to allow extensions to add a custom field to the 

search index used by Full-Text Search.
Also, as noted on the extension database models page, extensions can add custom fields either using extra_data or, whre the fields need to be indexable, custom models. Ideally, it should be possible to make both types searchable. Perhaps even using the index of a custom model directly.
By the way, the extension documentation on adding Review Request fields is an excellent starting point when it comes to adding custom fields that are stored in extra_data. It'd really help if it could include some starters on ones that use a custom model too!
What operating system are you using? What browser?

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