Can I extend the image field formatter without defining a custom one? XY problem: I need to extend the default image widget in the entity edit form with some custom fields so that, per image, e.g. a copyright and description, independently from the alt and title field, can be set.

I can add the fields by altering the core image module files, and by manually adding the respective database columns. But of course, I would like to alter the plugins with a custom module. 

I was able to create a custom formatter based on the ImageWidget as described in this tutorial.

My question is: is there a way to alter the default image widget/ field type to add custom fields rather than defining custom formatters/ widgets?
As an alternative, more flexible way, you could chose to use either File entity or Media entity, both give you a fieldable entity that you can add your additional fields, without having to write a single line of custom code.

File entity makes the files themself fieldable and adds file types, so images can have different fields than videos for example.

Media entity adds a separate entity that you work with, and works well combined with a number of other modules like Entity Browser and Inline Entity Form. There are some good tutorials out there:

http://www.annertech.com/blog/reusable-media-drupal-8-tutorial
http://flocondetoile.fr/blog/managing-media-drupal-8