At present, in Zimbra Desktop 2 beta onwards (and in Zimbra Web Client 6.? onwards), you can add custom fields. To do so, create or edit a contact. You will see several sections (email, phone, im, address, url, other, and notes).
In the 'other' section, there is a default field set as 'birthday', but you can also add an anniversary, and you can add 'Other' fields.
To do this, click on 'birthday' and select 'anniversary' or 'other' from the drop-down list. If you choose 'other' then you get to choose the label name of the new field yourself. You can add as many 'other' fields as you wish (in addition to a birthday and anniversaries).
Drawbacks:
1) Any fields you add will only appear above notes in the 'Other' section.
You cannot set new types of IM or Email or Phone numbers to appear in those relevant sections.
(Read/vote for enhancement here:
Bug 46074 – Custom type for Address Book fields )
2) You can only add new fields per contact, so if you want to have a new field which appears for all contacts, you have to add it manually for each one!
(Read/vote for enhancement here:
https://bugzilla.zimbra.com/show_bug.cgi?id=41745 )
3) You cannot set the data type for the field (it is just plain text) and the UI is confusing and looks like it's just for adding birthdays!
(Read/vote for enhancement here:
https://bugzilla.zimbra.com/show_bug.cgi?id=45993 )