But maybe the notion is "there's (not yet) an algorithm to judge a general point" - and that reminds me of the halting problem in computer science: No general agorithm exists, but there might very well be one that could prove e.g. that any C++ compiler with no more than 1000 lines of code halts on every input.

Yes, the conjecture is "there is no halting algorithm that you input a single c and it tells you in M-set or not". c is assumed "computable" meaning there is an algorithm that computes digit after digit of c. For if c itself is uncomputable (like Chaitin's number) it's trivial. The discussion in Penrose's book is in context of Goedel theorem and Turing halting problem. So the idea is there is no better way than to iterate and hope it either converges or diverges but if neither it may stay bounded forever or not, no way to know.

Of course there is a better way, using human creativity. As trivial example if c is in main cardioid there is a better way than to iterate forever and give up: just prove it's inside the cardioid which has a simple computable shape. For pretty much any non-escaping orbit you compute you can figure out in which cardioid or pseudo-circle blob it is and then prove it's inside by solving a polynomial equation which is computable. So the non-computable points must be on the boundary. Penrose can't give an example of a non computable point. At least that's my summary of the book chapter, I hope I got it all right; the chapter is written as "computability for dummies" so I hope I'm not worse than dumb.

The paper Claude pointed at defines "computability" in a more productive way AFAICU and seems to show M-set is computable in the sense you can make more and more accurate pictures of it and you can actually give guarantees on accuracy. But you already know that and are doing it.

Would be nice to come up with an algorithm (like an infinite series) that calculates a c such that no-one can figure out it it's in M or not. Proving uncomputability is an open problem so that would be too much to ask. Something like the first trancendental numbers that were proven; they were defined with an infinite series constructed explicitly so you can prove trancendentality.

Maybe pauldelbrot can come up with such a thing, at least I've never been able to ask a math question here that he could not answer