Here is the high-level overview of what you need to do, and I will suggest you to go through the code for existing fields to see how it works:
1) You need to add the HTML for that new input field that you want to add.
The HTML for the user details form is within the profile.php file, so just edit that file and add one more text input field there. Make sure that you set the "name" attribute for it since you will use that in the next step. For our example, let's say that you added "date_of_birth" as a name attribute for this field
The file that you need to edit is "assets/js/app/profile.js" and you'll need to extend the handler for "#update_details" button click which starts on line 44. All you need to do is to add one more item into the "details" object. The key there should be the name of your database column (let's say you also named it "date_of_birth") and the value is the value for your newly added form element.
Here is an example:
date_of_birth: form['date_of_birth'].value, // <-- this is what we added
And that's really it, it will then be sent to the ASAjax.php file and trigger the "updateDetails" method from ASUser class, which will then update the data in the database.