In Breaking Bad Walter White is known to be the master chef of blue meth, while yielding unheard of 99.9% purity. Is this possible or it is a big myth from the beginning?
Yes, it's possible.
In theory, it's possible to produce a 100% pure sample of any substance; if you isolated and removed any molecules that were not those of your target chemical. The trick is that it's usually very impractical to do so, for various reasons. The cost of purifying chemicals rises sharply as the purity goes up, because you need more expensive and sensitive equipment, or more complex reactants, etc. If you want the details, you'd need to go to a chemistry textbook (or perhaps try the Chemistry Stack, though I doubt they're going to explain the process of making methamphetamine to you :)
Having said that, it's completely possible for someone to produce near-100% pure crystal meth, because it's happened:
That crystal meth was later seized by law enforcement, tested and found to be more than 99 percent pure — purer than the infamously high-quality meth cooked by Walter White, the fictional teacher-turned-drug-lord in the popular TV series "Breaking Bad" src
Methamphetamine (aka Crystal Meth) has a specific chemical compound; C10H15N, as well as a known weight (149.2328 g/mol) and mass (149.120449 g/mol). Therefore, it can be produced with 99.9% purity if all other compounds are prevented from entering the formula, and the known weight and mass are achieved.
You may recall that Walt and Jesse used an extermination service as a cover-up at one point, and basically built a sterilized environment inside the houses. This was done to get the cook as close as possible to pure. If you had the time, resources and a controlled environment, you could most definitely produce a completely pure batch.
It is very easy for a skilled chemist to produce pure methamphetamine hydrochloride, a salt, because it crystalizes. A crystal will be very pure if grown properly. The challenge is to make sure the growth solution is pure to begin with by using pure components. If this is the case then the crystals will reject any foreign matter as they grown, making the result very pure, possibly with contaminants measured in parts per billion. The crystals must then be separated from the host liquor using proper washing techniques.
It is easy to produce very pure methamphetamine if you are a good chemist.
The reason why it is notable that Walter White does so is because most meth cooks are rank amateurs using bad recipes, poor equipment and little care. Meth isn't particularly hard to make (if you have the equipment and the right ingredients). It is also easy to make badly using poor equipment, bad recipes, little skill and readily available sources (like the over the counter medicine pseudo ephedrine).
Walter White is a good chemist and has access to the right ingredients and the right equipment. And he cares about his recipe and his professional standards. So it is entirely unsurprising that he can make very pure meth.
BTW this has been discussed in chemistry.SE: