Is it possible to show the Biographical Info textfield while creating a new user? I know it's possible to add user bio by editing a user AFTER it has been created. But I would like to do this while creating the user.
No. There are no hooks or filters to add an input field to the create user form.
Maybe it is possible to add an input field via jQuery. I have not tested it.
If it is pssible to add an input field, than it should be possible to save this information because the process of creating an user is the same as updating an existing user.
Yes, it is possible to add an input field with jQuery. And yes, it is possible to add extra data to the user creation.
As long as you use the same fields which are on the edit user screen, it's only a hand full lines of code:
In this example I just copy the markup from the edit user screen. So there is no need to do anything else because, as i mentioned above, creating an user is the same process as editing an user.
If you wish to create your own field (e.g. a field for a link to a twitter profile), than you have to hook into the edit users data and show users data. There are a lot of very good examples and tutorials on the net about this topic. One you can finde at WP Engineers.