JPATH_BASE already holds a correct path, for an administrator and for a site.
If you need access to administrator module then do request to /administrator.

Duplicate #25684.

Using client ID would be fine except that we can't determine it based on module name. Modules with the same name can be installed to both backend and frontend, e.g. like core mod_feed or mod_custom modules.

I do not think it is an issue, you just have to do a correct request:

for site modules: /index.php?option=com_ajax&module=foobar&format=json
for admin modules: /administrator/index.php?option=com_ajax&module=foobar&format=json

Same as with any component.

This is no good if you want to use Ajax in module form.

why you ever need that? :)

It the same as for components, you cannot execute Site component by doing request to Backend, and vice versa. The same applies to the modules.

To update form fields using AJAX. It's a reasonable use case.

Ah okay, I finally get it, it for a module configuration form, not for a module itself ?

Well, then, just add &client=0 to query, and update com_ajax to use it.
So you can call site module from backend. Eg:
/administrator/index.php?option=com_ajax&module=foobar&format=json&client=0

But, it should be forbidden to call administrator module from the site. So if someone call &client=1 from the site then com_ajax should throw an error.

@Fedik i use this in admin panel modules
client="administrator"