As of Version 1.1 of Paragon Chat, users again have the ability to create, edit and share bases for their characters.
The main difference is that the old COH bases were SG (super group) functions while the new Player Bases are either Account-Wide (all your toons in an account can share it) or Character-Private bases.
Any base can be shared with any player by starting a Team, inviting the guest to the Team, go to a Base Portal, go into your base, and ask the guest to choose that base at the Base Portal. Once inside, the guest can be booted from Team and others invited, up to the maximum of 32 characters at once (including your own characters).
Bases are started by visiting the Base Registrar (the former SG Registrar) in Atlas Park City Hall or the equivalent Red-side registrar and following the prompts.
GOOD NEWS FOR FORMER COH BASE BUILDERS! If you have a COH Demorecord taken inside your base from the old game, the Registrar has the ability to recreate the original SG base form that recording! YOUR BASE CAN LIVE AGAIN!
Details on Bases from the COHTITAN Forum:
Pictures from Player Bases: