[FEATURE] Feature Request: Auto-save conversation history locally

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

It would be useful to have a setting that automatically saves all conversations as local text files for later reference.
Claude Code VS Code extension

Proposed Solution

This could be a configurable option in settings specifying the output directory and format.

Alternative Solutions

_No response_

Priority

Medium - Would be very helpful

Feature Category

Configuration and settings

Use Case Example

_No response_

Additional Context

_No response_