Embodiments of systems, apparatuses, and methods for dual complex number by complex conjugate multiplication in a processor are described. For example, execution circuitry executes a decoded instruction to multiplex data values from a plurality of packed data element positions in the first and second packed data source operands to at least one multiplier circuit, the first and second packed data source operands including a plurality of pairs complex numbers, each pair of complex numbers including data values at shared packed data element positions in the first and second packed data source operands; calculate a real part and an imaginary part of a product of a first complex number and a complex conjugate of a second complex number; and store the real result to a first packed data element position in the destination operand and store the imaginary result to a second packed data element position in the destination operand.