There *are* anonymous chatbot or AI searches; lumo from Proton: https://lumo.proton.me/u/3/ and duckduckgo: https://duckduckgo.com/?q=DuckDuckGo+AI+Chat&ia=chat&duckai=1&atb=v406-1 which says: Your privacy is protected. All user chats are completely anonymous, and no identifiable information is recorded. The system ensures that your interactions remain confidential and secure. I trust these two, but of course, you make your own choice. I have found them very useful, though.