I'm just weaseling out because I don't want to claim that every system that can draw Mandelbrots is Turing complete (not considering pathetic cases like "draw_mandelbrot" keywords/functions/parameters).
Genuinely Turing-complete machines have infinite memory ("tape") and an arbitrarily large number of steps within which to complete an algorithm. I suppose "sufficiently Turing-complete" to mean that you can express an algorithm to the machine such that the algorithm can complete within the resources (RAM and time) that you have to give it.
I'll leave the proof to interested readers.