We analyse the computational complexity of the recently proposed ideal semantics within abstract argumentation frameworks. It is shown that while typically less tractable than credulous admissibility semantics, the natural decision problems arising with this extension-based model can, perhaps surprisingly, be decided more efficiently than sceptical admissibility semantics. In particular the task of finding the unique maximal ideal extension is easier than that of deciding if a given argument is accepted under the sceptical semantics. We provide efficient algorithmic approaches for the class of bipartite argumentation frameworks. Finally we present a number of technical results which offer strong indications that typical problems in ideal argumentation are complete for the class P^NP_‖ : languages decidable by polynomial time algorithms allowed to make non-adaptive queries to an np oracle.