Alexander,
Maximizer 9 contains an upgrade routine for making old databases usable with the newer versions. All you need to do is make a copy of the database and place it on the machine in a location you can easily remember (I would use C:\MaxData\[Database Name] as my location).
Then start Maximizer, hit cancel when asked to log in and cancel again when asked to open and address book. Then click File>>New Address Book, give it a name and point the location at the old 5.5 files. Then click ok.
The system may want you to log in, if so provide the username as master and the password as that databases master password.
It will then tell you something like "Maximizer files have been found in this location, click yes to use them for your new database or cancel to choose another location." at this point click yes.
It should now run through the upgrade process and open the database in Maximizer 9.