API is intended to be statless. And as such no you can't have access to session data (auth user).
Couple ways though.
Token based. A token is stored locally in JavaScript session. Then if available it's attached to the header (just like the csrf token in default install).
Draw back is each request where needed checks the token, queries the user and sets the user for that request cycle. Though default auth queries too.
Secondly you can set a meta and or script to allow user data to be grabbed. Similar to the csrf token again on the layout page.
Draw back is code lookers will easily see the data.
Thirdly you can respond with session data about the user on each request.
Drawback is keeping data in sync and code sniffers again.
The first one is considered best practice by majority of devs.