Hopefully the new window.crypto stuff could be used to createa a hosted webmail service where the private key is generated in the browser and never leaves the browser.
Probably not. Maddeningly, the W3C Web Crypto project decided to define a crypto interface in terms of primitives knitted together with Javascript, so, while you can probably assume WebCrypto AES is real native AES (assuming you're not dealing with polyfills, which is a real problem for any crypto extension), you can't assume the glue code in the cryptosystem is secure --- that's left up to content-controlled Javascript to define.