My husband has alopecia universalis. He lost all his hair at 16. He was on 60mg fluoxetine for a few years and it made no difference to him at all. He quite often gets periods of regrowth, but it never lasts and is itchy and horrible for him. He's been on no meds, different meds and now on venlafaxine.
If you need some more advice then talk to the lovely people at Alopecia UK - Amy is one of the founders and is truly lovely. They have a great FB group too.. https://www.facebook.com/groups/AlopeciaUK/ - It's honestly one of the nicest groups on FB!

I've not heard of fluoxetine causing alopecia as it's your immune system attacking your hair follicles, but I'm not a doctor...