Either I'm confused or the article is. Article seems to keep relating higher complexity to higher organization. But higher complexity is tied to higher entropy and disorganization, not higher organization. Generally, more complexity equals more entropy and less organization. Outside of certain possible exceptions like life (which still obeys it through waste product), is this not true?
It certainly seems to be that way in code, anyway.
Interpret entropy as "how unlikely is this state to occur by random chance?", with lower entropy corresponding to less likelihood.
If you throw a bunch of carbon, oxygen, and other organic compounds into a vat, you're vastly less likely to get a fully formed human being than, well, a disorganized pile of matter. Usually you wouldn't consider the random clump of molecules more complex -- there are practically an infinite number ways to rearrange a pile into a pile. In contrast, there are many orders of magnitude fewer ways for a human being to emerge.
That said, there is a sense in which the random pile of molecules is more complex. If I want to predict the motion of every (or at least, most) of the molecules in that vat, it's easier to predict the bulk motion of a human than each molecule moving per the Maxwell–Boltzmann distribution.
Similarly, it's easier to compress a static black image than continuous random noise. Notice that a static black image is more organized, more easily compressed, and less likely to occur by random chance (assuming noise is the default). Therefore, a static black image has less entropy. Whether you want to call this state more or less complex is a matter of definition.
No, the article is very explicit. In the traditional statements of the second law, higher entropy is indeed seen as lower organization.
However, the article is stating that this is not an accurate view of the mathematics - and that a more accurate view merely requires more disorganized distribution of energy, while matter may be arbitrarily better organized and still have its entropy increase. For example, crystalization of a material does not necessarily decrease its entropy, as long as the crystal formation dissipates heat.
I find this unconvincing. If the energy in your organized matter is not itself organized, the matter is not going to be moving in an organized way, which means it's dead.
The second law only applies to thermodynamic entropy, which has to do with the variety of ways that energy can be distributed among microstates (like maybe this electron is excited to 2 eV and that one is excited 3 eV) for a given microstate description (like "the sample is in the liquid phase").
It's tempting to simplify the universe like any other system: one for which microstate descriptions will have a fixed number of microstates across which the available energy will eventually be distributed more or less evenly. But perhaps that's a bad simplification. The article says as much, but in the middle of a bunch of claims about order and disorder.
So what's confusing about the article is it's attempt to describe the... confusion, that the new interpretation leads us away from.
It certainly seems to be that way in code, anyway.