[FEATURE] Feature request: Sync ~/.claude/ (global CLAUDE.md + memory) across machines

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

Problem
Global CLAUDE.md and auto-memory (~/.claude/projects/.../memory/) are stored locally in ~/.claude/. When working on multiple development machines, all user-defined coding style rules, learned preferences, and accumulated memory are lost on each new machine.

Use case
As a developer, I configure global coding style rules in ~/.claude/CLAUDE.md (e.g. Google docstrings, type annotations, vectorize with numpy). Claude also learns my preferences over time via auto-memory (e.g. "keep why-comments, remove what-comments"). When I switch to another dev machine or a remote container, I have to start from scratch.

Suggestion
Provide a built-in sync mechanism (e.g. tied to Anthropic account) so that global instructions and memory persist across devices automatically. This would make the Claude Code experience consistent regardless of which machine the user is on.

Proposed Solution

Account-based cloud sync

Sync ~/.claude/CLAUDE.md and ~/.claude/projects/.../memory/ to Anthropic account
Auto pull on login, auto push on change
Simplest user experience

Alternative Solutions

_No response_

Priority

Critical - Blocking my work

Feature Category

CLI commands and flags

Use Case Example

_No response_

Additional Context

_No response_