Memdex is a Chrome extension designed to enhance the user experience of AI conversations by saving and managing discussions across various platforms like ChatGPT, Claude, and Gemini. It addresses common issues such as difficulty in finding past conversations, the siloed nature of different AI tools, and the repetitive need to re-explain context. With Memdex, users can save conversations locally, detect relevant context in real-time, and seamlessly continue discussions without losing track of previous interactions.