<s>[INST] Defining the equation of an ellipse in the complex plane 
Usually the equation for an ellipse in the complex plane is defined as $\lvert z-a\rvert + \lvert z-b\rvert = c$ where $c>\lvert a-b\rvert$. If we start with a real ellipse, can we define it in the manner below?


For $x,y,h,k,a,b\in\mathbb{R}$ such that $a,b\neq 0$, we define a real ellipse as
$$
  \frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1.
$$
Let $z = \frac{x}{a} + i\frac{y}{b}$ and $z_0 = \frac{h}{a} + i\frac{k}{b}$. If we expand the equation for an ellipse, we have
$$
  \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{h^2}{a^2} + \frac{k^2}{b^2} -
  \frac{2xh}{a^2} - \frac{2yk}{b^2} = 1.
$$
Notice that $\lvert z\rvert^2 = \frac{x^2}{a^2} + \frac{y^2}{b^2}$ and $\lvert z_0\rvert^2 = \frac{h^2}{a^2} + \frac{k^2}{b^2}$. Now, let's write the ellipse as
$$
  \lvert z\rvert^2 + \lvert z_0\rvert^2 - \frac{2xh}{a^2} - \frac{2yk}{b^2} +
  \frac{yh}{ab}i - \frac{yh}{ab}i + \frac{xk}{ab}i - \frac{xk}{ab}i =
  \lvert z\rvert^2 + \lvert z_0\rvert^2 - \bar{z}z_0 - z\bar{z}_0 = 1.
$$
Thus, the equation of an ellipse in the complex plane is
$$
  (z - z_0)(\bar{z} - \bar{z}_0) = \lvert z - z_0\rvert^2 = 1\Rightarrow
  \lvert z - z_0\rvert = 1
$$
where $z$ and $z_0$ are defined above.
 [/INST] Setting $z_0=h+ik$, you can't do much better than
$$\frac{\left(\Re(z-z_0)\right)^2}{a^2}+\frac{\left(\Im(z-z_0)\right)^2}{b^2}=1.$$
The function in the LHS isn't analytic, so taking the real and imaginary parts isn't worse nor better than using the modulus.
You could also try starting from the parametric form $(h+a\cos \theta,k+b\sin\theta)$ with the hope to match it to the polar form $r\cos\theta+ir\sin\theta$ but you will end up with an anisotropic transform $(x,y)\to(\lambda x,\mu y)$, that doesn't have a nice form by complex functions.
The classical polar form
$$r=\frac p{1-e\cos\theta}$$
leads to the unsatisfactory equation
$$|z-z_0|=e\,\Re(z-z_0)+p,$$
that can easily be related to the first.
In all above cases, these representations are missing a degree of freedom (axis orientation).
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