Quizartinib With Azacitidine or Cytarabine in Treating Patients With Relapsed or Refractory Acute Myeloid Leukemia or Myelodysplastic Syndrome

Inclusion Criteria: * PHASE I * Refractory or relapsed disease defined as follows: patients with MDS or chronic myelomonocytic leukemia (CMML) should have failed prior therapy (e.g., with a hypomethylating agent, clofarabine, and/or with lenalidomide); patients with AML should have failed any prior induction therapy or have relapsed after prior therapy; patients (any age) with MDS or CMML who received therapy with a hypomethylating agent and progress to AML are eligible at the time of diagnosis of AML regardless any prior therapy for AML. The World Health Organization (WHO) classification will be used for AML; patients with any of the eligible diagnoses who have received no prior therapy are eligible if not candidates to receive standard intensive therapy (i.e., high-dose cytarabine-based chemotherapy). * Patients are eligible regardless of their FLT3 mutation status. * PHASE II * COHORT 2A: Patients with MDS, CMML or AML who are either: age 60 years or older and newly diagnosed, previously untreated. Prior therapy with hydroxyurea or single agent ara-C for the purpose of control of white blood cells (WBC) is acceptable.; age 18 years or older and with refractory or relapse disease who have received no more than one prior treatment regimen and will be receiving first salvage. For this purpose, a second induction cycle with the same drugs used during the first cycle, consolidation chemotherapy or stem cell transplant in complete remission (CR) (or complete response with incomplete platelet recovery \[CRp\] or complete response with incomplete bone marrow recovery \[CRi\]) will be considered part of the prior regimen. Prior therapy for MDS (or other malignancies) is not considered a prior regimen for AML in patients who progress from MDS (or other malignancies). * COHORT 2A: Patients (any age) with MDS or CMML who received therapy with a hypomethylating agent and progress to AML are eligible at the time of diagnosis of AML regardless any prior therapy for AML. The WHO classification will be used for AML. * COHORT 2A: Patients must have evidence of FLT3 ITD in their most recent assessment. * COHORT 2B: Patients with MDS, CMML or AML who are either: Age 60 years or older and newly diagnosed, previously untreated. Prior therapy with hydroxyurea or single agent ara-C for the purpose of control of WBC is acceptable or age 18 years or older and with refractory or relapse disease who have received no more than two prior treatment regimens and will be receiving second salvage, or who have received a prior SCT and will be receiving their first salvage. For this purposes, a second induction cycle with the same drugs used during the first cycle, consolidation chemotherapy or stem cell transplant in CR (or CRp or CRi) will be considered part of the prior regimen. Prior therapy for MDS (or other malignancies) is not considered a prior regimen for AML in patients who progress from MDS (or other malignancies) * COHORT 2B: Patients (any age) with MDS or CMML who received therapy with a hypomethylating agent and progress to AML are eligible at the time of diagnosis of AML regardless any prior therapy for AML. The WHO classification will be used for AML * COHORT 2B: Patients must have no evidence of FLT3 mutations in their most recent assessment * PHASE I AND II * Eastern Cooperative Oncology Group (ECOG) performance status =\< 2. * Bilirubin =\< 2 x upper limit of normal (ULN). * Alanine aminotransferase (ALT) =\< 2.5 x ULN. * For patients with suspected liver infiltration from leukemia ALT should be =\< 5 ULN. * Creatinine =\< 2 x ULN. * Serum potassium, magnesium, and calcium (normalized for albumin) levels should be at least within institutional normal limits. * Patients must provide written informed consent. * Patients must have been off chemotherapy for 2 weeks prior to entering this study, unless there is evidence of rapidly progressive disease, and must have recovered from the toxic effects of that therapy to at least grade 1. Use of hydroxyurea for patients with rapidly proliferative disease is allowed before the start of study therapy and for the first four weeks on therapy. The additional days of hydrea after 28 is permitted as clinically indicated, on case by case basis after discussion with the principal investigator (PI). Other agents given transiently with the intention to control rapid proliferation such as 1-2 doses of single agent ara-C or few doses of sorafenib are also allowed. * Women of childbearing potential must practice contraception. Women considered not of childbearing potential include any of the following: no menses for at least 2 years or menses within 2 years but amenorrheic for at least 2 months and luteinizing hormone (LH) and follicular stimulating hormone (FSH) values within normal range (according to definition of postmenopausal for laboratory used) or bilateral oophorectomy or radiation castration and amenorrheic for at least 3 months. Females of childbearing potential should practice effective methods of contraception. Effective methods of contraception include barrier methods (e.g., condoms, diaphragm), spermicidal jelly or foam, oral, depo provera, or injectable contraceptives, intrauterine devices, tubal ligation, and abstinence. Male patients with female partners who are of childbearing potential should also practice contraception. * Negative urine or serum pregnancy test. Exclusion Criteria: * Patients with known allergy or hypersensitivity to quizartinib, mannitol, AZA, cytarabine or any of their components. * Serum potassium \< 3.5 mEq/L despite supplementation, or \> 5.5 mEq/L. * Serum magnesium above or below the institutional normal limit despite adequate management. * Serum calcium (corrected for albumin levels) above or below institutional normal limit despite adequate management. * Patients with known significant impairment of gastrointestinal (GI) function or GI disease that may significantly alter the absorption of quizartinib. * Patients with any other known disease concurrent severe and/or uncontrolled medical condition (e.g. uncontrolled diabetes, cardiovascular disease including congestive heart failure, myocardial infarction within 6 months and poorly controlled hypertension, chronic renal disease, or active uncontrolled infection) which could compromise participation in the study. 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 have active disease or be in remission for less than 6 months. * 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. * Patients with known malignant disease of the central nervous system. * 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) at screening and on day 5 prior to the first dose of AC220. The QTcF will be derived from the average QTcF in triplicate; if QTcF \> 450 msec on day 5, AC220 will not be given; patients with congenital long QT syndrome; history or presence of 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); sustained heart rate of \< 50/minute on pre-entry ECG; 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; patients who require treatment with concomitant drugs that prolong QT/QTc interval or strong CYP3A4 inhibitors or inducers with the exception of antibiotics, antifungals, and antivirals that are used as standard of care to prevent or treat infections and other such drugs that are considered absolutely essential for the care of the subject. * Known family history of congenital long QT syndrome.