How are $z$ and $z^*$ independent? I have been told that a complex number $z$ and its conjugate $z^*$ are independent. Part of me understands this, since for two independent variables $x$ and $y$ we can always define new independent variables $x' = \alpha x + \beta y$ and $y' = \alpha x - \beta y$.
However, this contradiction is confusing me:
Suppose I assume $x$ and $y$ are real. Then if I know $z$, I know both $x$ and $y$, which sort of makes sense because $\mathbb C \cong \mathbb R^2$. For example, if you tell me $z = 4 + 5i$, then $z^*$ is uniquely determined to be $4 - 5i$. How can we then say $z$ and $z^*$ are independent? I cannot change $z$ without also changing $z^*$. I can, however, change $x$ without changing $y$.

In several comments, OP mentions the quantity $\frac{dz^*}{dz}$, so I think it would be useful to clear something up.
If $f:\mathbb{R}\rightarrow \mathbb{R}$ is a function on the real numbers, then we can define its derivative (assuming it exists) as
$$ f'(x) = \lim_{h \rightarrow 0} \frac{f(x+h) - f(x)}{h}$$
Recall that for the limit to be well-defined, we must arrive at the same value whether $h\rightarrow 0$ from the left or from the right.

If $f:\mathbb{C}\rightarrow \mathbb{C}$ is a function on the complex numbers, then we similarily define its derivative as
$$ f'(z) = \lim_{h \rightarrow 0} \frac{f(z+h)-f(z)}{h} $$
Notice that there is a subtle difference here.  Now $h$ is complex, so there are an infinity of directions along which it could potentially go to zero.  For this limit to be well-defined, we must arrive at the same value for all of these cases.
Applying this to the complex conjugate map,
$$ \frac{dz^*}{dz} = \lim_{h\rightarrow 0}\frac{(z+h)^* - z^*}{h} = \lim_{h\rightarrow 0} \frac{h^*}{h}$$
The issue becomes clear when we write $h=|h|e^{i\theta}$ and $ h^*=|h|e^{-i\theta}$.  The derivative becomes
$$\frac{dz^*}{dz} =  e^{-2i\theta}$$
If $h$ is purely real ($\theta=0$), then $\frac{dz^*}{dz}=1$.  If $h$ is purely imaginary ($\theta=\pi/2$), then $\frac{dz^*}{dz}=-1$.  Obviously the value of the derivative of the complex conjugate map depends on the direction of the infinitesimal displacement $dz$.  This is no good - it means that $\frac{dz^*}{dz}$ is not well-defined, and so $z^*$ is not a complex-differentiable function of $z$.
It follows from this that if a complex map is a function of both $z$ and $z^*$ (say, $f(z)= \mathbb{Re}(z) = \frac{1}{2}(z+z^*)$), then it is only complex-differentiable if you consider it a function of two independent variables and differentiate accordingly.  In other words, we can say that
$$ f = f(z,z^*)$$
$$\frac{\partial f}{\partial z} = \frac{1}{2}, \frac{\partial f}{\partial z^*} = \frac{1}{2}$$
but $\frac{df}{dz}$ is not a meaningful notion.
