Piwik could perfectly track internal websites searches.
Making tracked search key words a link in visitor log (like all tracked pages are)
would make it a breeze to see what the visitor has seen when searching.
This would help to understand why he act in the way he does e.g. why he choose this strange site from search results..
Probably the settings for search query and category in
Administration/websites/Tracking Internal Site Search
has to be slightly modified or detailed to make Piwik know the exact structure for links to search
at this time only "q" is set as query parameter and "cat" is set as category parameter
one need in addition: