Built-in browser pane: session doesn't persist, no URL history/autosuggest

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

The built-in browser pane in the Claude desktop app (Code tab) loses its session/tab after sitting idle for a while — e.g. a signed-in Google Sheets tab disappears and has to be re-navigated and re-authenticated. There's also no memory of previously visited URLs, so there's no autosuggest/history when typing a new URL.

Both of these make multi-turn work in the browser pane (e.g. having Claude read/reference the same external page across a long session) more friction than it needs to be.

Proposed Solution

Persist the browser pane's session/tabs for longer (or indefinitely while the Claude Code session is active), and add URL history with autosuggest when navigating — similar to how Codex's browser behaves.

Alternative Solutions

Codex's browser already does both of these (persistent sessions, URL history/autosuggest) — worth matching that behavior.

Priority

Medium - Would be very helpful

Feature Category

CLI commands and flags

Use Case Example

I had a Google Sheet open in the browser pane, signed in, and was working through a multi-turn conversation with Claude to build a financial plan referencing that sheet. After some time between turns, the pane's session dropped and the sheet had to be re-navigated and re-authenticated to continue — interrupting a task that spanned many chat turns. Separately, when starting a new browser task I have to retype full URLs from scratch since there's no history/autosuggest, even for sites I visit constantly in these sessions.

Additional Context

_No response_