[FEATURE] Support conversation timestamps in "Claude Code for VS Code"

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

I'd like to know when my query and CLUDE CODE’s answer happened, but there is no timestamp displayed in the "Claude Code for VS Code".

Proposed Solution

Add timestamps in conversational interface of the "Claude Code for VS Code"
This helps me trace when these things happened. For example, when did I raise the issue and when did claude code solve it

Alternative Solutions

_No response_

Priority

Medium - Would be very helpful

Feature Category

Interactive mode (TUI)

Use Case Example

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Additional Context

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