Would you really SSH through this to your server? I mean, what about security...
It'll let you run whatever terminal app you like (defaults to SSH w/advanced features like session duplication).
Full disclosure: I am the author of Gate One.
A possible workaround is implementing SSH agent forwarding support in Chrome app. However, you still have to trust remote ssh binary to only do what it's supposed to do while you're connected (i.e. not log your communications, not open secondary channels doing some weird stuff and so on).
Yet in your own site you say: "Stuck on a Windows machine and can’t stand using PuTTY to SSH into work? "
I have to agree with OP, using this for any SSH work would be asking for trouble even if one were to use SSH keys with passwords.