This research investigates the recursive sequence given by $X_{n+2}=imX_{n+1}+X_n$, where $X_1=X_2=1+i$, with $i=\sqrt{-1}$ and $m$ being a parameter that assumes real values. We analyze the recursive sequence as a second-order difference equation with constant coefficients. By solving this equation, we derive explicit bounds on the parameter $m$ and obtain general equations to describe the resulting geometric shapes. Our analysis reveals how said bounds on $m$ influence the behavior of the sequence. We plot the sequence in the complex plane and as $m$ varies, we can see interesting geometric shapes formed from conic sections. When $m$ lies in the interval (-2,2), the sequence exhibits bounded behavior with points in the complex plane tracing two ellipses. For $ m >2$, the sequence becomes unbounded, leading to hyperbolic trajectories. This work explores the geometric nature of the solutions, examining how the sequence’s behavior evolves as the parameter $m$ varies.