Try this, if I've explained it right it will work.
Let's say that the field you want notification of updates to is called "email" and you want to know when someone updates their email. Create two new fields called "new email" and "old email". When the update occurs, the user inputs the new email into this field. Hide or remove the original "email" field from the form as it won't be relevant. Rename the "new email" field's label on the form to appear as though it is the original "email" if necessary.
Then, when they edit the form and input the "new email", have the record rules do the following in this order:
update field "old email" to "email"
update field "email" form value "new email"
update field "new email" to "" (blank)
Then on submit rules, when old email is not blank and old email does not match email then send a custom email to email@example.com
Then create a daily task for the table called old email patrol or something and have it update "old email" to record value "email" when "old email" does not match "email". Make a second task updating "new email" to "" (nothing-blank) on the same basis.
If you need old email to match email immediately, maybe because the same form might be used to change other fields later on and you don't want it to send you an email update notification each time another field is updated until the patrol comes along, create a new yes/no field called "Confirm" with default being yes and create a new child page on submit on the form you are updating the email on called "confirm changes" create a list for the record and remove everything except for old email and email and name the label for old email say "Your email has been changed from" - then display "old email" then change label for "email" to "to" then display "email". Then display an edit form for the record only showing the [Confirm? "y/n"] and the submit button. Then, on this form, in record rules; if Confirm = No - change email to old email and if confirm = Yes update old email to match email. In submit rules if confirm = No send custom email notifying that email has been changed again, in this case changed to the email it was just changed from, ultimately generating two emails notifying of email updates, one to a new email and another update back to the old one. It should be rare that the user will change their mind, but even so, they technically have changed their email twice anyway. Also in submit rules, set If Confirm = Yes display message "changes have been saved" or something.
The same process is applicable even if your change is made by updating a connected record. But you might need to use Text Combo fields in the form you are updating them from in order to measure the changes and create the triggers. If the field is updated externally, using Zapier for example, then just get Zapier to send the update. If the fields are changed using the API, then you probably wouldn't need my help with this because your code would be notifying you of the changes.
Alternatively, if you ARE using Zapier, you could just get Zapier to send the update, Trigger: When field updated Action: Send email using GMail or SMTP or Zapier Email or something. I recommend Zapier for everything Knack. EXCEPT WORKING OUT HOW TO EFFING SELECT A USER BASED ON A FIELD VALUE FROM ANOTHER SERVICE.SDJKLAbdflbjhasdgjl