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The dynamics of the family of maps $\displaystyle{f_{\alpha, \beta, \gamma,\delta}(z)=\frac{\alpha z + \beta}{\gamma z^2 +\delta z}}$ in complex plane isinvestigated computationally. This dynamical system $z_{n+1}=f_{\alpha, \beta,\gamma, \delta}(z_n)=\frac{\alpha z_n + \beta}{\gamma z_n^2 +\delta z_n}$ hasperiodic solutions with higher periods which was absent in the real linescenario. It is also found that there are chaotic fractal and non-fractal likesolutions of the dynamical systems. A few special cases of parameters are alsohave been taken care.

Computational Complex Dynamics of $\displaystyle{f_{\alpha, \beta, \gamma, \delta}(z)=\frac{\alpha z + \beta}{\gamma z^2 +\delta z}}$