Last modified field for user profile?

As mentioned in the comments, you can use the Plugin and Metadata APIs to attach some functionality to the 'profile_update' action such that whenever a user’s profile information receives an update, custom user-metadata is set to the time of the update:

function wpse216609_update_profile_modified( $user_id ) {
  update_user_meta( $user_id, 'wpse216609_profile_updated', current_time( 'mysql' ) );
}

add_action( 'profile_update', 'wpse216609_update_profile_modified' );