This paper extends the notion of a \( p \) -hyponormal operator to bounded right linear quaternionic operators defined on right quaternionic Hilbert spaces. Several fundamental properties known for complex \( p \) -hyponormal operators are investigated in the quaternionic setting. To develop these results, we establish the Furuta inequality for quaternionic positive operators. This inequality provides the foundation for discussing the \( p \) -hyponormality of quaternionic operators and their Aluthge transforms. Finally, we introduce a new class of quaternionic operators lying between quaternionic \( p \) -hyponormal and quaternionic paranormal operators.
