Yes, your Koi will "live" on trout chow, catfish chow, or other types of food. I even met a person who fed ground up cat food, not catfish food, CAT FOOD. Likewise, you and I could live on Donuts, or bacon cheeseburgers, but we wouldn't be in the best of health.
If you expect your koi to grow healthy, strong, & disease resistant, make sure you feed a healthy "Koi" food. I'm not going to endorse one food over another here, because there are many quality foods out there and they keep getting better. I just ask that you read the labels, compare brands, & buy food designed specially for koi. It is even a good idea to use more than one so you are sure to get enough of the proper vitamins.
You have probably heard not to use food that has "corn" as one of the first 3 ingredients because Koi can't digest it. But did you know what the following deficiencies can do?
Vitamin A Impaired growth, body depigmentation, pop eye.
Vitamin D3 Poor growth, impaired calcium balance.
Vitamin E Anemia, increased water retention.
Vitamins C
Scoliosis, deformative growth, hemorrhagic skin,
liver, kidney, intestine, and muscle.
Riboflavin Cloudy eyes, hemorrhagic eyes, dark body coloration.
Niacin Loss of appetite, jerky or difficult motion.
Menadione Prolonged blood clotting, anemia.
D-pantothenic Loss of appetite, clubbed gills, gill exudate.
Folic acid Lethargy, fragile caudal (tail) fin.
Pyridoxine
Hyper-irritability, loss of appetite, rapid and
gasping breathing.
Thiamine Poor appetite, convulsions, loss of equilibrium.
Biotin Spastic convulsions, skin lesions.
B 12 Poor appetite, anemia.
Calcium Reduced growth.
Manganese Loss of equilibrium, dwarfism, high mortality.
Zinc Anorexia, fin and skin erosion.
Iodine
Thyroid hyperplasia (abnormal increase of thyroid
cell growth).
Copper
Lack of important metabolic enzymes.