Per 2018 International Residential Code, I believe that any dedicated bedroom that you construct has to have its' own egress window but you need to check with your local building inspectors to determine how they interpret the code. They might allow the bedroom to be constructed elsewhere.
I like
@Midnighter 's baravarian pub idea, but that space with the window might be the only place you can locate a dedicated bedroom, unless you go to the expense of adding another egress window in the bedroom space ($$$). Unfortunately, if you close off that recessed area with a wall+door for the bedroom, you will effectively lose natural light to the remainder of the basement unless you install welled-windows elsewhere.
Egress window requirements ensure that homes meet the International Residential Code for fire safety exits in basement level living spaces. Read more here.
www.egresswindows.com
www.proremodeler.com