square of differential operator

Can differential operators be squared algebraically in the usual way? What would this even mean? Is this just the differential operator applied twice consecutively?

For example, does it make sense to compute $\displaystyle (\frac{\partial}{\partial x} - i \frac{\partial}{\partial y})^2 = (\frac{\partial}{\partial x})^2 -2i\frac{\partial}{\partial x}\frac{\partial}{\partial y}-(\frac{\partial}{\partial y})^2$? If so, what is to be done with, say, $\displaystyle (\frac{\partial}{\partial x})^2$? Is this just $\displaystyle \frac{\partial ^2}{\partial x^2}$?