Purpose : Hide the Search Select Advanced Field. It is easy to hide the text fields(using Capt_/Data_), but to hide the gif image / underlying URL the following script can be used.

This script is tested on crm57 version for the new case webpicker window. may be you try the script similar to the one below. sourceIndex-80/82/ is valid in CRM 5.7J only, it may differ in other versions)

You can basically put an alert to find the source index and hide the url as well , in addition to hiding the gif image.

for(i=0; i -1)
{
document.all[i].style.visibility = hidden;
}
if(document.all[i].src.search(/smallmenuselect.gif/i) > -1)
{
document.all[i].style.visibility = hidden;
}
}
}