This phase I/II trial studies the side effects and how well cladribine, idarubicin, cytarabine, and quizartinib work in treating patients with acute myeloid leukemia or high-risk myelodysplastic syndrome that is newly diagnosed, has come back (relapsed), or does not respond to treatment (refractory). Drugs used in chemotherapy, such as cladribine, idarubicin, and cytarabine, work in different ways to stop the growth of cancer cells, either by killing the cells, by stopping them from dividing, or by stopping them from spreading. Quizartinib may stop the growth of cancer cells by blocking some of the enzymes needed for cell growth. Giving quizartinib with cladribine, idarubicin, and cytarabine may help to control acute myeloid leukemia or high-risk myelodysplastic syndrome.