In a 2d space, in which $Distance=\sqrt{x^2+y^2}$, with $x$ being a real number, and $y$ being a real multiple of $a+bi$, with $a$ and $b$ both being real none zero constants, would it be possible to have a continuous curve in which all points are the same none zero distance from a specific point?
I understand that in the case of $a$ as a none zero real number and $b$ as $0$, or $a$ as $0$ and $b$ as a none zero real number that it is possible to have a continuous curve with every point being the same distance from a certain specific point, with the former being a euclidean circle, and the later being a hyperbola, but I'm not sure about the case of $a$ and $b$ both being none zero real numbers?