[FEATURE] VSCode extension: add a "Background Tasks" panel (parity with Desktop app)

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 Claude Code Desktop app has a background tasks side pane showing all running subagents/workflows. The VSCode extension has no equivalent — /workflows only surfaces workflow-tool runs inline in chat, and background subagents (launched via the Agent tool) only show up as one-shot completion notifications once they're already done. While a subagent is actively running, there's no way to see:

• All currently running/queued subagents at a glance
• What each subagent is actively doing right now (live status, not just a final summary after it finishes)
• The live message/tool-call stream of a subagent while it's in progress
• Scheduled/cron routines and /loop timers, which today are only visible from the CLI via claude agents — not from inside the IDE at all

This makes it hard to trust or supervise background work from the editor. I have to either wait for a completion notification with no insight into progress, or drop out of VSCode into a terminal to run claude agents just to check if something scheduled is still alive.

Proposed Solution

Add a "Background Tasks" panel/view to the VSCode extension, matching what the Desktop app already provides:

• A dedicated sidebar or bottom-panel view (similar to the Output/Terminal panel pattern) listing every active piece of background work: subagents spawned via the Agent tool, Workflow tool runs, /loop dynamic-mode timers, and cron-scheduled routines — all in one list, not scattered across /workflows output and CLI-only claude agents.
• Each entry shows live status (running/queued/completed/failed), a short description of what it's doing, and elapsed time.
• Clicking an entry drills into its live message/tool-call stream, so I can watch a subagent's progress in real time instead of only seeing the final summary.
• The panel updates live as tasks start, progress, and finish — no need to reopen or refresh.
• Scheduled/cron routines appear here too, so I don't need the CLI just to check whether a background routine ran or is still pending.

Alternative Solutions

Currently I work around this by waiting for the one-shot completion notification in chat, with zero visibility while the subagent is actually running. For scheduled/cron work I have to switch to a terminal and run claude agents from the CLI, which breaks the IDE-only workflow. The Desktop app already solves this with its background tasks side pane, so the VSCode extension is the outlier.

Priority

High - Significant impact on productivity

Feature Category

Interactive mode (TUI)

Use Case Example

Example scenario:

1. I'm working on a multi-module project and kick off two subagents in parallel — one reviewing code, one running tests — plus a scheduled /loop check that polls a deploy status every few minutes.
2. While they run, I want to glance at a panel in VSCode and see all three: what each is doing right now, not just wait for them to finish silently.
3. With this feature, I could open the Background Tasks panel, see the code-review agent is "reading file X," the test agent is "running suite Y," and the loop task is "waiting, next check in 4m."
4. This would save me from either flying blind until a completion notification arrives, or breaking flow by switching to a terminal to run claude agents.

Additional Context

• Reference: Claude Code Desktop app's existing background tasks side pane — this request is asking for parity with that, inside the VSCode extension.

• Would also be valuable for cron-scheduled routines (CronCreate) to show next-run time and last-run result in the same panel.