[Feature Request] Chrome extension conversations not saved to Claude chat history and lack persistent memory across sessions

_This report was generated by Claude Sonnet 4.6 from the Chrome extension._

Feedback from a Chrome extension browser session (Claude Sonnet 4.6)

During a browser automation session today, I helped a user explore the Chrome extension's capabilities — including archiving emails in a demo inbox, creating a Google Calendar event, and booking a haircut at a local Zurich barbershop with no public API.

Two limitations came up that I'd flag as meaningful gaps:

1. Browser extension sessions don't appear in Claude chat history.

The user checked and confirmed this conversation was not saved to their claude.ai history. This makes browser sessions fully ephemeral — useful work, discovered URLs, and context are lost when the window closes. Syncing extension conversations to chat history (even optionally) would make the tool significantly more useful for iterative or recurring tasks.

1. No persistent memory across browser sessions.

The user asked whether I'd remember their barber's name and booking URL in a future session. I wouldn't. This contrasts with desktop Claude's memory feature and creates friction for exactly the kind of recurring task the extension excels at (booking appointments, checking calendars, etc.). Bringing memory support to the extension — or at minimum allowing shortcuts to carry context — would close this gap.
The user worked around both limitations by creating a saved shortcut with the direct booking URL embedded. That's a good partial solution, but native memory and session history would be meaningfully better.

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Session info: Claude Sonnet 4.6 · Chrome Extension (browser control beta) · macOS · May 11, 2026 · Sites accessed: claude.ai, calendar.google.com, thebarberparadox.ch, thebarberparadox.hairlist.ch