Douady rabbit

The Douady rabbit is a Julia set (http://planetmath.org/SetDeJulia) produced by

$$c=-\frac{1}{8}+\frac{3}{4}i.$$

As the Mandelbrot set indicates, the real part can be varied to as much as $-.2$ while the imaginary part can be varied to as much as $.75i$ and still produce a connected set.