The admin email that is used as the email address for notifications etc does not handle incoming emails to be able to see them in WordPress (Although this is an interesting concept).
You need to have the email address setup elsewhere, with your host for example, and then anyone that replies to an email from there you can see in whatever email client you use.
I think your best bet would be simply to use your gmail address as the admin notification email address, as well as the email address for your administration user.
Let me know if that wasn’t clear enough.
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