This is not about email monitoring services, but instead if you are looking for powerful hosted or web email clients with IMAP support and multiple accounts support, then here is a selection of products that you can consider.
AfterLogic is an email server, email client and also offer email components for businesses.
Looking for web based IMAP email client
If you are looking for web based IMAP email client you should know that there are some free services around.
Gmail Google Email
You may already know about Google Gmail service, of course using Gmail you can configure multiple accounts (see note below) and you can also use IMAP accounts.
Zoho email is one of the favorite alternatives with IMAP support and web based solution. You can sign in with your Google account and start using it for free. In order to add imap account in Zoho mail you first need to enable IMAP access in settings, then go to account management and add a new account.
AIM Mail client
For other email clients, you can consider desktop applications like Thunderbird or Outlook. However, notice that if you want to avoid to backup your hard disk to keep your old emails you may consider using IMAP to keep messages on the server.
Unfortunately, at the moment of writing this post there is no useful information about cloud email clients, but clearly services like Hotmail or Gmail offer email in the cloud. However, businessmen and corporations that use Google for email management would find there is a lack of support of multiple accounts in a professional way. Although you can define multiple accounts, at the time you send emails it adds a message that says Behalf of your original account, so if you use your personal account for Gmail and try to define multiple other accounts you’ll face that your message sent to coworkers or colleagues will appear with a “behalf” label.
Mail Smith is another alternative for email client software.
If you are curious about IMAP ports, you should know that 143 is the default IMAP port, while other servers use different port settings.