Genetically Modified T-cell Immunotherapy in Treating Patients With Relapsed/Refractory Acute Myeloid Leukemia and Persistent/Recurrent Blastic Plasmacytoid Dendritic Cell Neoplasm

Inclusion Criteria: * ARM 1 - AML: Research patients enrolled are those patients with relapsed or refractory CD123+ AML de novo, or secondary OR participants who are at high risk for disease recurrence NOTE: CD123+ biphenotypic acute leukemia or CD123+ acute lymphoblastic leukemia (ALL) may also be considered but only after discussion with the study principal investigator (PI) * Relapsed AML is defined as patients that had a first complete remission (CR) before developing recurrent disease (increased bone marrow blasts) * Refractory AML is defined as patients that have not achieved a first CR after 2 cycles of induction chemotherapy; for patients with AML evolving from myelodysplastic syndrome, they should have completed at least one cycle of induction chemotherapy * ARM 2 - BPDCN: Research participants with a diagnosis of BPDCN, according to World Health Organization (WHO) classification by hematopathology, who underwent at least 1 line of systemic therapy for BPDCN and who have persistent or recurrent disease in at least one of the following are eligible: peripheral blood, bone marrow, lymph nodes, spleen, cutaneous lesions or other sites OR participant who are at high risk for disease recurrence * FOR BOTH STUDY ARMS: Research participants must have bone marrow and/or peripheral blood samples available for confirmation of diagnosis of AML or BPDCN; CD123 positivity must be confirmed by either flow cytometry or immunohistochemistry within 90 days of study entry; cytog…