Hi Mads,
It is! In the alert email you are sending yourself, click the token button at the top in the email editor (lbrown chalkboard with quotes on it).
Then search for the fields you want to include in the email. They could be, first name, last name, email address, and anything else. The tokens look like this:
{{lead.First Name:default=edit me}}
{{lead.Last Name:default=edit me}}
{{lead.Email Address:default=edit me}}