complex geometry – Is there any way to tell if any givin Number will either blow up or tend towards a Value when squared?

Ok so as a Set Up, i want to do the all time classy, the Mandelbrot set. As far as i know, the equation

$$Z rightarrow Z²+C$$

Just asks the question “Ok givin any Complex Number, if you Square the Non Complex terme and then Add the Complex compnent, is this new number something REALLY huge or does it tend towards a Value ?

Of course, normaly this Equation is based on Iterations so you do that a bunch of times and bam, either your Value is huge for small.

But, there is sort of a Problem for me. At the moment, my code spits out this:Image

I dont want to be a downer but that doest really look right. My guess is that the Problem is how huge the values get once you square them a few times. And since Computers are pretty dumb, at some point Values might switch from Positive to negative and so on.

So, to the question, please tell me there is a fancy Algorithm that can just tell me if Z will be huge or small D:

Thanks !