The answer in my experience is yes but how you do it depends on your email configuration.
I may be misinterpreting your post but in your first sentence you say you are dealing with one email address but later you imply you are using 2 address.
If indeed you are wanting incoming mail to go to two computers, then the following will work with Outlook Express.
Both computers will have to be configured for the same email account. Then, from the top menu go to Tools/Accounts/Properties/Advanced and tick the "Leave a copy of messages on server" tickbox. You may wish to select the option to delete from server after a number od days. You will need to do this on both your computers. This will cause incoming messages to be uploaded to both machines.
Outlook has the same option under Tools/Email Accounts........More settings.
The above works for me with a desktop and a laptop using a POP3 email account.
There is another solution which requires you to set up one computer as the email server and the other computer accesses the server's inbox but I haven't tried that as my first solution does the job perfectly for me. It would have the advantage of being able to access your SENT box from either computer. It can be a bit iritating trying to find a sent email and then remember it was sent from the other computer.