Say I have a function on the Cartesian plane for which I have all its pairs $(x, f(x))$.
How can I graph this function as imaginary? Would it be possible to take all its pairs, say one pair of $f(x)$ is (2,3), and change that to $a + bi \to 2 + 3i$? Is that too "random" to do, or is there a better way?
Is it also correct to assume that the x axis is the Real part, and the y axis is the imaginary, as shown in this link for Rectangular notation? Link.
There is a "more proper" way to think about it, but there is no unique way to do it. Starting with your real function $f$, you want to extend this to a complex function. So you need to find $f(z)$ for complex numbers $z$. Writing $z$ as $a+b\mathfrak{i}$ this is the same as asking what $f(a,b)$ is where you only know the values of $f(a,0)$. There is not a unique way to fill in the missing values for $b \neq 0$. For example look at the the two function $g(a,b) = 2a+3ab\mathfrak{i}$ and $h(a,b)=2a-4b\mathfrak{i}$. Both of these functions have the same real values, but are different as complex functions.
As for you question about the axis, the real axis is usually drawn horizontally, and the imaginary axis is drawn vertically, but you shouldn't really call them the $x$- and $y$-axis anymore.