A noninvertible, two-dimensional, discrete-time system featuring multistability is presented. Because the preimage behavior of this system is a function of location in phase space, the boundary separating the basins of attraction can be disconnected. These "polka-dot" basins of attraction have either a finite number of preimages (giving a finitely-complicated basin) or infinitely many (giving infinite complexity). A complexity criterion based on following the noninvertible region forward in time is presented and a fixed-point algorithm for computing the boundary of the "complete" noninvertible region is discussed.