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 limits * Ability to take oral medication * Ability to understand and provide signed informed consent * Baseline test of left ventricular ejection fraction \>= 50% * Women of childbearing potential (WOCBP) must have a negative serum or urine pregnancy test within 7 days * WOCBP must use appropriate method(s) of contraception such as oral contraceptive pills (OCP), birth control shots, intrauterine device (IUD) etc. WOCBP should use an adequate method to avoid pregnancy until 30 days after the last dose of investigational drug. Men must agree not to father a child and agree to use a condom if his partner is of child bearing potential. Women who are not of childbearing potential (ie, who are postmenopausal or surgically sterile) as well as men with known azoospermia do not require contraception * Patients with isolated extramedullary myeloid neoplasm will be eligible Exclusion Criteria: * Any coexisting medical condition that in the judgment of the treating physician is likely to interfere with study procedures or results * Breastfeeding women * Patients with current active malignancies or any remission for \< 6 months, except patients with carcinoma in situ or with non-melanoma skin cancer who may be in remission for less than 6 months or have active disease * Active clinically serious and uncontrolled infection. Patients with recent infections must have no temperature of \>= 101 degrees Fahrenheit (F) for at least 48 hours (hrs) (before first dose, day 1) * Patients with known significant impairment of gastrointestinal (GI) function or GI disease that may significantly alter the absorption of quizartinib * Documented active central nervous system leukemia (patients with history of central nervous system \[CNS\] leukemia without active disease are allowed) * Patients with a known confirmed diagnosis of human immunodeficiency virus (HIV) infection or active viral hepatitis * Patients who have had any major surgical procedure within 14 days of day 1 * Impaired cardiac function including any of the following: * Screening electrocardiography (ECG) with a corrected QT (QTc) \> 450 msec. The QTc interval will be calculated by Fridericia's correction factor (QTcF). The QTcF will be derived from the average QTcF in triplicate. Patients are excluded if they have QTcF \>= 450. Subjects with prolonged QTcF interval in the setting of RBBB (right bundle branch block) may participate upon review and approval by the medical monitor. RBBB for patients' triplicate electrocardiograms (EKGs) can show false QTc prolongation; therefore, the cardiology collaborator for this study will manually review to provide an accurate reading of the QTc * Patients with congenital long QT syndrome * Sustained ventricular tachycardia requiring medical intervention * Any history of clinically significant ventricular fibrillation or torsades de pointes * Known history of second or third degree heart block (may be eligible if the patient currently has a pacemaker) * Heart rate of \< 50/minute on pre-entry ECG * Left bundle branch block * Right bundle branch block + left anterior hemiblock (bifascicular block) * Patients with myocardial infarction or unstable angina within 6 months prior to starting study drug * Congestive heart failure (CHF) New York (NY) Heart Association class III or IV * Atrial fibrillation documented within 2 weeks prior to first dose of study drug * Known family history of congenital long QT syndrome * Patients who are actively taking a strong CYP3A4 inducing medication