Why can't imaginary numbers be irrational?

If you defined irrational numbers as $\mathbb{C}\setminus \mathbb{Q}$ rather than $\mathbb{R}\setminus \mathbb{Q}$, then you would be in the uncomfortable position of calling both $i+1$ and $\sqrt{2}+\pi i$ irrational, even though the first looks almost like a rational, even an integer, whereas the second looks more like what we expect from an irrational.
Instead, it's cleaner to define Gaussian rationals as those complex numbers $a+bi$ where both $a$ and $b$ are rational. So the first example above is a Gaussian rational (in fact a Gaussian integer), whereas the second is not.