"Imaginary" and "real" numbers are so-called because at one time, it was thought that real numbers exist and imaginary ones don't. We now know that this is not true, but the labels "real" and "imaginary" stuck around, which is unfortunate because they are misleading.

Imaginary and real numbers are both real in the sense that you can do maths with them. And they're both imaginary in the sense that they're concepts, not physical things.

It's also worth noting that imaginary numbers are not just a contrivance for taking square roots of negative numbers, they are a core part of mathematics with uses far too numerous to list.