We're working on Trezor-backed web password manager with the same feature. You'll be able to reach all of your passwords on any computer (with Trezor connected, of course), without need of typing any "master password" as is common in standard password managers (like Keepass).
You'll just choose which password do you want to unlock at the time, so you'll be able to log into any website on friend's computer without a risk of revealing internet banking credentials or other high profile passwords...
Edit: Password manager is of course needed for sites which do not offer challenge-response authentication, but I believe that some day at least bitcoin-related sites will implement password-less login into their site like GreenAddress already did: https://twitter.com/GreenAddress/status/479939415088062464