If you want to open Gmail’s compose window without seeing your entire inbox, here is the way of how to do it:

  1. Add this link to your Bookmarks bar

https://mail.google.com/mail/u/0/?view=cm&fs=1&tf=1

  1. Click the link

  2. Compose window will open without showing your inbox

  3. Write and send that email

Done!

A Gmail compose window opened on its own, with the Compose Mail button highlighted