You may play the card exiled with Compleated Conjurer even if Compleated Conjurer leaves the battlefield or you lose control of it.

Playing the card exiled with Compleated Conjurer follows the normal rules for playing that card. You must pay its costs, and you must follow all applicable timing rules. For example, if the card is a creature card, you can cast that card by paying its mana cost only during your main phase while the stack is empty.

Unless an effect allows you to play additional lands that turn, you can play a land card exiled with Compleated Conjurer only if you haven’t played a land yet that turn.

If you don’t play the card, it will remain exiled.