Store .claude.json backup files inside ~/.claude/ directory

Feature Request

The .claude.json.backup.* files are currently stored in the home directory root alongside .claude.json. This clutters the home directory with multiple timestamped backup files:

.claude.json
.claude.json.backup
.claude.json.backup.1771256537816
.claude.json.backup.1771256544993
.claude.json.backup.1771256548271
.claude.json.backup.1771256548309
.claude.json.backup.1771256548313

Proposed Change

Store backup files inside ~/.claude/ instead of the home directory root. The ~/.claude/ directory already exists for Claude Code configuration, memory, and settings — backups belong there too.

For example:

• ~/.claude/claude.json.backup (or ~/.claude/backups/)

Rationale

• Keeping the home directory clean is a common user expectation
• The .claude/ directory is the natural home for all Claude Code state files
• Hidden dotfiles in $HOME are typically limited to one directory or one config file per application, not a growing set of backups