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.

Adam Walter

Admin Notice is a very simple plugin that allows you to display custom notices in the WordPress admin that is visible to all users.
Control over the notice text and styling is available from the plugin’s settings page.
Developers can use the <code>admin_notice_enable</code> hook to control the notice’s visibility on a more granular level. The following example demonstrates how to show the notice only in the “Posts” area of the admin.
<pre><code>add_filter('admin_notice_enable', function($enabled) {

 $screen = get_current_screen();

 if (isset($screen->post_type) && $screen->post_type !== 'post') {
 $enabled = false;
 }

 return $enabled;

}, 10, 1);
</code></pre>

Admin Notice

Nice Plugin!

Display a custom notice to all users in the WordPress admin.

Comment Admin Notifier vs Admin Notice