One solution that works for me is to add both email addresses under the same contact in Gmail.
To do this, go to Gmail > Contacts, find the person (either email works if both already exist in the contact list), and click on Add Email right under the first email address already there. Then paste the second email for that person. Finally, make sure the full name is correct for you to be able to find it again later, when you're composing an email (sometimes, Gmail can't figure out the name of the person based on their email address only).
If both email addresses have never been used before in Gmail (and you haven't manually created the contact entry for that person), then you will have to create the contact yourself and enter both email addresses, with the full name of the person. If you already exchanged at least one email with the person, Gmail will have saved that email address, but again, beware it might lack the full name of the person which is important for you to be able to find it again later.
Bonus tip for the power-user: If you already have two contact entries for the same person, you can select both in the contact list, and click More > Merge Contacts. This will take both email addresses and put them in one contact entry, which is extremely useful. It also merges other infos, so if you created an entry for Bob with his phone number on your smartphone (assuming your Gmail contacts are synced), and have another entry for Bob Adams in Gmail, you can select both of them (knowing they are the same Bob) and merge and get one nice contact with all informations in one contact.