EDIT: Would it work for what you want to do to download your Family Tree as a Plain Text file, open it in a word processor or text editor, and search from there?
Below is my first, and entirely less help ful answer to your question:
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I can't think of a way to search for text that you entered into the "Bio notes" field, because there is not a way to display that information on the screen. Information you entered into Profession, Company, Interests, or Activities could be searched for by displaying that field on your Family Tree (under show options at the bottom) and then searching in the browser (generally CTRL-F on a PC and CMD-F on a Mac) for the info you are looking for.
Of course, the person who has that information has to be displayed currently on your Family Tree, and they probably have to actually be visible on the screen for you to see the text highlighted. But I only tried this in Chrome on a Mac. Other browsers might make this easier.
You could always View the Page Source, and search for the information there, but, again, this only works for fields that you can display on the screen.
You might want to use the Send Feedback button at the bottom of your family tree to request that the Bio Notes field also have an option to display on the screen for just this reason.
I realize this isn't a great answer, so I'm going to leave it as a comment instead of an answer in hopes that someone else will notice your unanswered question and leave a better one.
