Screen has both 'c' and 'C-c' set to new window by default. tmux only has 'c' by default.

From https://www.gnu.org/software/screen/manual/html_node/Default...:

  C-a c
  C-a C-c
  (screen)
  Create a new window with a shell and switch to that window. See Screen Command. 

Edit: d'oh. I had to reread the original comment. You already know this, but by adding 'C-c' I don't have to release the Ctrl button.

You can add this to tmux.conf to get screen-like behavior:

  # New Window
  bind C-c new-window