Yes it must be in at least one other place.

 I have managed to reconfigure everything to how I want it. I deleted my main account and recreated it. Now it checks all three pop3 accounts that I want with no @paradise showing up. I do have a 2nd identity called gmail (and somewhere there is a file that must have the @paradise embedded in it) but I have left check all identities off.

So somehow the original account had other stuff in it that you don't have direct access to and the only way to remove it was delete it and recreate it. In the beginning I didn't recreate it but only changed the original.

I have just changed my ISP (actually same ISP but different server). Previously I was name@paradise.net.nz. I changed to name@clear.net.nz. I changed all the settings under Internet options to reflect @clear instead of @paradise and email works fine EXCEPT something within Pegasus is still trying to access name@paradise.net.nz.

I am currently checking 3 POP accounts, the new one, gmail and a 3rd one. None of these appear to be using the old @paradise as far as I can tell. I have just disabled the two extra accounts and am supposedly only checking the @clear POP account but it still persists in also checking the old account.

Where is this old account coming from ?

Were you using multiple users or just multiple identities? The POP and SMTP connections are actually stored in *.pnd files. How many are present that begain with  POP? You can open them in Notepad to view their contents. One of those .pnd files has the old server and user name showing in it.

When you look at window in Tools, Internet options and receiving POP3 how many are shown. In the window (tab) before is block checked to check all?

I was using multiple identities but currently disable except for the default. There are three POP3 accounts in receiving and the box is checked to check all.

I checked the *.pnd files and found the culprit. It is the master email address for the old account. Probably got inserted somewhere 10 years ago.

1. Can I just delete the pnd ?

2.  Why wasn't this address showing up anywhere where I could see it ? I would have thought that if the box is checked to check all that somewhere handy I could "see all" address's that were being checked.

I would not delete that .pnd but just rename to .bak and test and then if all ok in a day or so go ahead  delete the thing. I like to keep a copy at least until after testing and later OS versions that make so easy by just copy and  paste of the file shows a copy done to same location.

The thing, old address, may be elsewhere in configuration but I did not mention every possible location, just where it was know to be and could be found.

I renamed the *.pnd file to .old but made no difference.

The default profile checks 2 of the 3 POP3 accounts. The 3rd account is a gmail account and it is this one which also checks the @paradise account. With the check all identities box unchecked the gmail pop3 isn't checked and therefore the incorrect @paradise also isn't checked.

I have removed the gmail identity but there appears to be no way to make the default identity check the POP3 for gmail which is showing  in the Receiving POP3 under Internet Mail Options.

Work continuing.....