Base of Atiesh

The Base of Atiesh, Greatstaff of the Guardian is a quest item and the only piece of Atiesh that could not be found within Naxxramas. It is available to players who have collected the forty from Naxxramas and who have formed the Frame of Atiesh, and received the quest from Anachronos to obtain both the base and the Staff Head of Atiesh. This item is needed to complete the quest for Atiesh, Greatstaff of the Guardian. Once the frame and base have been attained, only the head - in the hands of Kel'Thuzad - remains.

Source
The Base of Atiesh drops off C'Thun in the Temple of Ahn'Qiraj for players who have the quest Atiesh, the Befouled Greatstaff.