When creating a new window or pane in tmux the path will be set to ~/ by default. You can rebind the keys to make the new window or pane open with the current path:

# .tmux.conf
bind '"' split-window -c "#{pane_current_path}"
bind % split-window -h -c "#{pane_current_path}"
bind c new-window -c "#{pane_current_path}"

This will work in tmux >= 1.9.