It really depends on how you define $\mathbb{R}^2$, though you'll probably also have to exercise great care in how you define $\mathbb{C}$ as well. It also depends on what you mean by "okay." If by "okay" you mean "it fits well with widely accepted definitions of $\mathbb{R}$ and $\mathbb{C}$," I'd have to say that no, it's not okay.
Have you heard the one about the magician and the mathematician who were locked in a cage together? As soon as the door closed, the magician started tapping on the wall and listening to the sounds that made. "You're just going to sit there while I do all the work of getting us out?" the magician asked. "I already got us out," the mathematician replied, "but you were too busy knocking on the wall to notice." The mathematician had redefined "outside" to mean inside the cage.
The kernel of truth to that joke is that you can redefine anything.
But for a redefinition to have value, it has to pass two tests. First, it has to be consistent with all the other definitions you provide, and in this case, it looks like you have to provide a definition for $\mathbb{C}$ even though most of the time there would be no need to. Second, the redefinition must accomplish something, even if it's something small.
Every few months, someone decides that there is a fundamental problem with the Fibonacci sequence and that it is up to them to fix it. Sometimes their redefinition fixes the problem without seriously affecting the various well-known identities for Fibonacci numbers. But the problem their redefinition fixes is so small and inconsequential that it does not really justify any efforts to disseminate the new definition.
So the question you have to ask yourself is this: Does definiting $\mathbb{R}^2$ and $\mathbb{C}$ so that $\mathbb{C} =\mathbb{R}^2$ accomplish anything you consider worthwhile?