[FEATURE] Feature Request: Cloud-based conversation history for Claude 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

Feature Request

### Problem
Currently, Claude Code conversations are only stored locally, while claude.ai conversations are stored in the cloud. This creates an inconsistent user experience where users
cannot access their Claude Code conversation history across devices or view all their Claude interactions in one place.

Proposed Solution

### Proposed Solution
Add cloud-based conversation history for Claude Code that allows users to:

1. View Claude Code conversations in a web interface (similar to claude.ai)
2. Sync conversation history across devices
3. Optionally integrate with claude.ai conversation history for a unified experience

Alternative Solutions

_No response_

Priority

Critical - Blocking my work

Feature Category

CLI commands and flags

Use Case Example

### Use Case
As a developer using Claude Code on multiple machines, I want to access my previous conversations from any device and maintain a complete record of my interactions with
Claude.

Additional Context

Additional Context

This would improve the consistency between Claude's different interfaces and provide better user experience for developers who use Claude Code regularly.