Sean Hayes

Comments On is a plugin that adds a column to the “All Posts” or “All Pages” view in the Admin dashboard. The column will display one of three options for the related post/page: “open”, “closed” or “registered_only”.
Now, quickly and easily, without having to click “Quick Edit” or “Edit” for each post you can find out which posts/pages can accept comments and which can’t.
It is an ideal companion for the Comment count column alerting you to content that hasn’t any comments. Or it can highlight any posts/pages that might not be enabled for comments.

Comments On

Comments On adds a column to the "All Posts"/"All Pages" view in the Admin dashboard showing the status of comments.

Jordi Cabot

In the Settings->Discussion page, authors of a post can use the checkbox Email me whenever – Anyone posts a comment.
But this does not send an email as well to the site admins. In blogs where you have a number of guest authors, you may want to be informed about all the new comments so you can respond (if the author is missing) or just participate in the discussion.
As a site admin myself, I was missing many comments. This means plenty of missing opportunities to engage with your audience.
To solve this situation, the plugin adds a new checkbox in the Discussion page. If checked, admins will get an alert email for new comments.

Comment Admin Notifier

Very simple plugin that doesn't try to do many things. But the one it does is very useful: as a website owner you want to make sure you're on top of all the comments in the site even comments to posts you didn't write. 

It does one thing and it does it right

With this plugin, admin users get an email alert every time a new comment is posted on ANY post in the site.

Comments On vs Comment Admin Notifier