This phase I trial studies the side effects and best dose of modified immune cells (IL13Ralpha2 CAR T cells) after a chemotherapy conditioning regimen for the treatment of patients with stage IIIC or IV melanoma. The study agent is called IL13Ralpha2 CAR T cells. T cells are a special type of white blood cell (immune cells) that have the ability to kill tumor cells. The T cells are obtained from the patients own blood, grown in a laboratory, and modified by adding the IL13Ralpha2 CAR gene. The IL13Ralpha2 CAR gene is inserted into T cells with a virus called a lentivirus. The lentivirus allows cells to make the IL13Ralpha2 CAR protein. This CAR has been designed to bind to a protein on the surface of tumor cells called IL13Ralpha2. This study is being done to determine the dose at which the gene-modified immune cells are safe, how long the cells stay in the body, and if the cells are able to attack the cancer.