Cladribine, Idarubicin, Cytarabine, and Quizartinib in Treating Patients With Newly Diagnosed, Relapsed, or Refractory Acute Myeloid Leukemia or High-Risk Myelodysplastic Syndrome

Inclusion Criteria: * Diagnosis of * AML (World Health Organization \[WHO\] classification definition of \>= 20% blasts, excluding Acute promyelocytic leukemia), * Acute biphenotypic leukemia or * High-risk MDS (\> 10% bone marrow blasts) * Frontline cohort: Patients aged 18 to 65 years * Relapse cohort: Patients aged \>=18 years old * Patients may be newly diagnosed (Frontline cohort) or with prior therapy (Relapsed cohort) as follows: * For frontline cohort: Patients must be chemonaive, i.e., not have received any chemotherapy (except hydroxyurea \[Hydrea\] \[no dose limit\], tretinoin \[atra\] \[no dose limit\] or ara-C \[one or two doses (max 2 gr/m\^2 per dose)\] for transient control of hyperleukocytosis) for AML or MDS. They may have received hypomethylating agents for prior MDS and transfusions, hematopoietic growth factors or vitamins. Temporary prior measures such as apheresis or Hydrea are allowed * For relapsed cohort: Patients with previously treated, relapsed or refractory AML, acute biphenotypic leukemia or high-risk MDS (\> 10% bone marrow blasts) * Eastern Cooperative Oncology Group (ECOG) performance status =\< 2 * Creatinine \< 1.5 mg/dl * Total bilirubin \< 1.5 mg/dL, unless increase is due to hemolysis or congenital disorder * Transaminases (serum glutamate pyruvate transaminase \[SGPT\]) \< 2.5 x upper limit of normal (ULN) * Potassium, magnesium, and calcium (normalized for albumin) levels should be at least within institutional normal limi…