Show agent/session name on the left side of the input box

Feature Request

When using Claude Code, the agent/session name appears in the header area. It would be useful to also display it on the left side of the input box (next to the prompt), so users can always see which agent or session they're interacting with — especially when running multiple agents or sessions side by side.

Current behaviour

• Agent/session name is shown in the header bar only
• In full-screen terminal mode, the header can scroll out of view or be hidden

Proposed behaviour

• Show the agent/session name as a label on the left side of the input box (similar to how some shells show the current git branch or environment)
• This keeps the context always visible regardless of scroll position or terminal mode

Why it matters

• When working with multiple Claude Code sessions (e.g. in tmux panes or multiple terminal tabs), it's easy to lose track of which agent you're talking to
• The input box is always visible, making it the most reliable place for persistent context

Related

• Header disappears in full-screen mode (see separate issue)