This fractal consists in a combination (using complex division) of two Julia II fractals.

Conformal maps by Robert Bunney

A close up of 4 of the hundredth roots of unity. By Ian Sammis.

A plot of z = |W(x+iy)|, where W is the Lambert W function

The Real part of z^z^z

A plot of the complex valued function (z-2)²(z+1-2i)(z+2+2i)/z³ by Hans Lundmark

Real part of the j-invariant as a function of the nome q on the unit disk

This image by Stephen Schiller has been one of my favorites for years. It shows the distribution of fractions of Gaussian integers with a restriction on the denominator. Explicitly, this is the set of complex numbers (a + bi)/(c + di) where a,b,c, and d are integers, and √(c² + d²) < 25. Source: The Pattern Book, by Clifford Pickover.

"Visualisation of the (countable) field of algebraic numbers in the complex plane. Colours indicate the leading integer coefficient of the polynomial the number is a root of (red = 1 i.e. the algebraic integers, green = 2, blue = 3, yellow = 4...). Points becomes smaller as the other coefficients and number of terms in the polynomial become larger. View shows integers 0,1 and 2 at bottom right, +i near top."
(http://en.wikipedia.org/wiki/Algebraic_number)