[FEATURE] Claude in Chrome: Log session conversation, activity, etc in sidebar chats

Preflight Checklist

• [x] I have searched existing requests and this feature hasn't been requested yet
• [x] This is a single feature request (not multiple features)

Problem Statement

Claude in Chrome's sidebar chat currently provides no local or accessible record of the actions it takes on a user's behalf (navigation, clicks, form fills, etc.), nor the conversation had with the user. When a browsing session ends, there's no way to review what the extension did, audit its behavior, or recover context from a previous session.
This is especially important as the extension gains more powerful capabilities like long-running workflows and scheduled tasks — the more autonomously it operates, the more users need visibility into what happened.

Why this matters:

• Transparency / trust: Users delegating autonomous tasks need to be able to verify what was done
• Debugging: When something goes wrong mid-workflow, there's currently no trail to follow
• Privacy auditing: Users on Team/Enterprise plans in particular may need records for compliance purposes
• Context recovery: Being able to resume or reference a past browsing session is valuable for multi-step workflows

Proposed Solution

Provide some form of session activity log, which could take one or more of these forms:

• A local log file (similar to ~/.claude/chrome/chrome-native-host.log that already exists for debugging) containing a human-readable summary of actions taken per session
• Session history surfaced in the extension's side panel (e.g., "last session: visited 3 pages, clicked 2 buttons, filled 1 form")
• Integration with Claude.ai's existing chat history, so browser actions are recorded inline with the conversation that triggered them

Alternative Solutions

None. There is no supported way to retrieve a log of what Claude in Chrome did in a session.

Priority

Low - Nice to have

Feature Category

Other

Use Case Example

1. I'm using Claude in Chrome to research a topic — asking questions in the sidebar while Claude reads and navigates pages on my behalf
2. I close the browser or the session ends (tab closed, timeout, crash, etc.)
3. Later, I want to reference something Claude found or said during that session — either to share it with a colleague, cite it in a document, or pick up where I left off

With session logging, I could open a history view in the sidebar (or find a local log file) and retrieve the conversation and the actions Claude took. This would save me time because right now that context is simply gone — I have to start over or rely on my own notes, even though Claude did the work

Additional Context

_No response_