According to Next Web, the new Twitter feature to download your history is now active for some people.
It's not active yet for my account, but I'd still like to learn more about the format. So if it's available for your account, please consider sharing your archive with us techies so we can have a look inside.
If you have an archive, you can upload it somewhere or email it to dave dot winer at gmail dot com and I'll upload it so everyone can have a look. Thanks!
Update: Thanks to Navjot Singh, we now have an archive to explore.