A feasible model is introduced that manifests phenomena intrinsic to iterative complex analytical maps (such as the Mandelbrot set and Julia sets). The system is composed of two alternately excited coupled oscillators. The idea is based on a turn-by-turn transfer of the excitation from one subsystem to another [S.P. Kuznetsov, Example of a physical system with a hyperbolic attractor of the Smale-Williams type, Phys. Rev. Lett. 95 (2005) 144101] accompanied with appropriate non-linear transformation of the complex amplitude of the oscillations in the course of the process. Analytical and numerical studies are performed. Special attention is paid to an analysis of the violation of the applicability of the slow amplitude method with the decrease in the ratio of the period of the excitation transfer to the basic period of the oscillations. The main effect is the rotation of the Mandelbrot-like set in the complex parameter plane; one more effect is the destruction of subtle small-scale fractal structure of the set due to the presence of non-analytical terms in the complex amplitude equations.