This chapter addresses the algebraic formalization of paracomplete systems Pn* with quantifiers, which extend the propositional paracomplete calculi Pn developed by da Costa and Marconi. Paracomplete logic challenges the Principle of Excluded Middle, allowing both a proposition and its negation to be simultaneously false. The chapter introduces the syntax, axioms, and semantic principles underlying the Pn* systems, emphasizing their duality to paraconsistent calculi and their alignment with vagueness and fuzzy reasoning. A special emphasis is given to the development of P1*-monadic algebras, which serve as algebraic semantics for the quantified version of P1. These structures generalize Curry algebras by incorporating quantifier-like operators under a paracomplete framework. The chapter presents representation theorems relating P1*-monadic algebras to Boolean-valued functional algebras and explores their connection with traditional monadic algebras via quotient constructions. Open problems regarding the classification and uniqueness of associated monadic algebras are also discussed, highlighting the structural richness of these paracomplete algebraic systems.