How to re-create a notifications 'inbox' similar to the one here at stackexchange?
You could use the Menu Badges module for such notifications, which you can add to any menu item and/or menu tab of your choice. In your case something like "New comments" seems like one, out of a few, appropriate menu item (or menu tab).

Here are some details about it (from its project page):


  Provides a method for adding iOS-style badges to menu items. (see screenshot)
  
  
  
  Adding Badges to Menus
  
  Once enabled, go to Administration &gt; Structure &gt; Menus, and click "list links" next to the menu containing the target item. Click "edit" next to the item and select the badge to display with the Display Menu Badge select box.
  
  Adding Badges to Tabs
  
  For adding badges to menu tabs, there is a tab under Administration &gt; Structure &gt; Menus called "Tab Menu Badges". Search for the menu router path of the tab you want to alter, then select a badge from the corresponding select box.
  
  Badge Types
  
  The module includes five example badges, and new badges can be created easily with Views. Modules can also supply their own badges with a hook implementation. See the README.txt file for details.


Because of how it integrates with the Views module, by using this integration you basically reduce the solution for any type of notification to something like "Just create an appropriate view of it".

However, there is a small caveat: if you really want the actual number (shown in the red circle) to be accurate, make sure you have some field in your view that will actually correspond to the "new" part of your question. To better explain this: have a look at the "tracker" view (disabled by default), which has the field "Content:Has new content" (or variation: "Content:Has new comments"). This is a perfect field to use if you want to use Menu Badges to get notified whenever there is new content (or variation: new comments).

If you don't have such "new" indicator, then you basically have 2 variations to get around that:


You give the user the option to selectively mark those entries so that they are no longer included in the counter. It takes a little effort to implement such option, but the tutorial about "How to allow users to manage their own Message Stack messages?" explains how to use the Flag, Rules and VBO modules to get it to work. 
When the user visits the menu link (or menu tab), you "mark" all of the entries included in the Menu Badges counter as "no longer new". That should make the counter disappear entirely. The anser to "How to make menu badges count disappear after the page is viewed?" details the steps how this can be implemented.


Video tutorials:


Defining Menu Badges with Views in Drupal, from the module maintainer, which shows how to use it for a custom view with a menu tab "unpublished content" within admin/content.
Drupal 7 Menu Badges Module.