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.

Simon Wheatley

This plugin is designed for administrators to share the task of moderating comments and publishing posts marked as “Pending Review”. It is NOT designed to notify non-admin subscribers of new comments or posts.
The plugin adds checkboxes to the WordPress profile page, allowing you to setup notifications for these events:
<ul>
<li>a comment is added</li>
<li>a comment is held pending approval</li>
<li>a post is pending review, </li>
<li>a post is published</li>
</ul>
If you have permission to edit other users, you can also setup notifications for them… so when you go on holiday, you can set a colleague to receive the comment notifications while you’re away.
Users can only receive notifications for comments if they have the authority to moderate comments, and can only receive post notifications if they have the authority to publish posts.
Plugin initially produced on behalf of <a href="http://twentysixlondon.com/" rel="nofollow ugc">Twenty Six</a>.
Is this plugin lacking a feature you want? I’m happy to discuss ideas, or to accept offers of feature sponsorship: <a href="http://www.simonwheatley.co.uk/contact-me/" rel="nofollow ugc">contact me</a> and we can have a chat.
Any issues: <a href="http://www.simonwheatley.co.uk/contact-me/" rel="nofollow ugc">contact me</a>.

Email Alerts

Allows users to configure whether they wish to be alerted when a comment is added, comment held, post posted.

Comment Admin Notifier vs Email Alerts