[FEATURE]: display timestamps on all conversation messages (UI/context) #41589

Checklist

• I have verified this feature has not been suggested before (searched existing issues before filing).

Not a duplicate of #29399 / #35348 (checked before filing):

• [FEATURE]: TUI: timestamp gutter mode for messages #29399 (closed, has a linked PR): TUI gutter mode showing HH:MM per message — a pure display-mode feature.
• [FEATURE]: Show message/run start and completion timestamps with millisecond precision #35348: message/run start & completion timestamps with millisecond precision — pure display feature focused on run lifecycle, not per-message content.
  This request goes beyond pure display gating: it asks for timestamps that (a) are shown per conversation message (user + assistant + summary), (b) are timezone-configurable with an on/off toggle, AND (c) crucially — a mode where the timestamp is visible to the LLM context too, so the model itself can reason about the temporal order of a multi-day session. The existing two issues only cover display, not LLM-perceivable time. Optionally verified in DB so plugin-verification is possible.

Feature

Add a timestamp (e.g. [HH:MM:SS] East-Asia time, configurable) to every conversation message shown in the TUI and/or context: user messages, assistant messages, and summary messages.

Why (reason)

Today none of the conversation messages carry a timestamp. In long sessions it is impossible to tell when each message/summary happened — the user cannot trace the timeline of a multi-day session, what happened before/after, or correlate events in time. This is a basic traceability need for real-world usage.

The user already tried to implement it as a plugin (timestamp_plugin.js), which injects [HH:MM:SS] into part.text on the message.part.updated hook and filters it out of speech broadcasts. But the plugin's self-logged "injection succeeded" is not trustworthy: the injected prefix never actually reached either the stored DB text or the LLM context — it only polluted the TUI display intermittently. So a native/robust first-class implementation is required.

Request details

1. Timestamps should be display-only (TUI rendering) and optionally usable in context; should NOT pollute the persisted message text / LLM context unless the user opts in.
2. Timezone configurable (many users are not in UTC).
3. A toggle so it can be turned on/off.

Please iterate on this soon — it is currently blocking a clean long-session experience.