Feature request: Make it visible whether a session runs locally or remotely (session list indicator)

Problem

In the Claude Code desktop app it is hard to tell at a glance whether a session in the session list is running locally on my machine or remotely in a cloud environment. The session titles and the list view look identical in both cases, so I have to open a session or inspect its details to find out where it is actually executing.

This distinction matters in practice: the machines involved do not have the same files. A file, repository, or network share that exists on my local workstation may simply not be present in a remote environment (and vice versa). When I pick up an existing session, I need to know which machine it runs on before I ask it to work with local paths, mapped drives, or company-internal resources.

Suggested solution

• Show a clear visual indicator (badge, icon, or label such as "Local" / "Remote") next to each session in the session list.
• Ideally, also show the host name or environment name, so users with several machines can tell them apart.
• Make this information filterable or sortable in the session overview.

Workaround

As a workaround I currently prefix my session titles manually with "L" (local) or "R" (remote), but a built-in indicator would be much more reliable.

Environment

• Claude Code desktop app on Windows 11 Pro