Answering Other's Questions About Oneself
tl;dr: I think it is normal, you just need time to get used to it. It will pass. I think it will be useful for you to investigate, why do you feel the desire to be "normal"? What part of you feels insecure?

I can only offer my anecdotal support - yes, I've felt like this too. 

At one point I couldn't even fill out the "About me" section of my social media profile, or a form for a retreat that asked me to describe myself. I mean, what do I put there? I couldn't really define myself by my samsaric interests of the past, and I didn't identify as a Buddhist or some other "ist" either because it's my present vehicle to the truth, it is not me. 

Then I began to wonder if my friends will consider me weird. And I also realized it didn't matter much to me what anyone thought.

It took me a while to accept totally that this identity-less person is who I am, and I am fine with it. What helped me was to listen to several interviews of spiritually awakening people on batgap.com - after hearing several hundred interviews I had the confidence that this is quite okay. I was not very different from the people being interviewed in my mind state and experiences, and they all seemed pretty together in their external identities after a few years of being lost, so I felt okay.

These days if someone asks me who I am, I don't have the deer in the headlights look - instead I try to imagine what the person wants to hear, and reveal that aspect of my externally visible identity, even though internally I have none.

And yes, being okay with dying is pretty normal at this stage of acceptance.