s₀=0.3, s_i+1=(-0.5+0.25i)*s_i²+(1.2+0.5i)
Center:0+0i; Zoom = 0.5

Some more playing around with complex parameters in the a*s_i²+b sequence generator.

I have not investigated the long term behavior of this sequence. Given the large capture set and Mandlebrot-like right side, I suspect it the sequence converges to very short cycle, most likely a single point.

A couple of zooms to follow.