The moon of Proxima Centauri b would appear to go through a full cycle of phases as seen from the parts of the surface of Proxima Centauri b where it was visible.
And the moon of Proxima Centauri b would probably have to be much more massive than Earth's Moon, and orbit much closer to Proxima Centauri b, to have a stable orbit.
The Hill sphere or Hill radius is the distance within which a planet can maintain satellites in stable orbits.
I found an online Hill Sphere calculator.
I set the planetary orbit's semi-major axis at 0.04856 AU, and the eccentricity at 0.01, since it seems to be unknown whether Proxima Centauri b has any orbital eccentricity.
I set the mass of Proxima Centauri at 0.1221 the mass of the Sun and the mass of Proxima Centarui b at 1.07 the mass of Earth.
That gives a Hill radius of 453,778 kilometers for Proxima Centauri b. However:
The Hill sphere is only an approximation, and other forces (such as radiation pressure or the Yarkovsky effect) can eventually perturb an object out of the sphere. This third object should also be of small enough mass that it introduces no additional complications through its own gravity. Detailed numerical calculations show that orbits at or just within the Hill sphere are not stable in the long term; it appears that stable satellite orbits exist only inside 1/2 to 1/3 of the Hill radius.
Thus a long term stable orbit for a moon of Proxima Centauri b would be within about 151,259 to 226,889 kilometers of the planet.
As well know, the Moon originated much closer to Earth than it is now. Tidal interactions between the Moon and the Earth slowed down the rotation of the Earth and slowed the rotation of the Moon until it was tidally locked to the Earth and also pushed the Moon father and father from Earth. at the present time the semi-major axis of the Moon's orbit is 384,399 kilometers.
So if your hypothetical moon originated at a similar distance from Proxima Centauri b as the Moon did, and has a similar mass to the Moon, it should have been pushed out to a similar distance from Proxima Cantauri b during the time span of the age of the Moon. And a distance close to 384,399 kilometers would be far beyond the true region of stability of Proxima Centauri b.
Thus it would have been quite likely that the moon would have been perturbed out of orbit around Proxima Centuri b. Since Proxima Centauri is a bit older than the Sun, the hypothetical moon should have had plenty of time to be forced out of orbit around Proxima Centauri b.
So you may need to change, the age, the mass, or the initial formation distance of the hypothetical moon of Proxima Centauri b. I note that making the formation of the moon more recent involves a big problem. If the moon is a normal moon, it should have formed about the same time as Proxima Centauri b, and thus about the same time as Proxima Centauri, and thus be significantly older than Earth's Moon.
But Earth's Moon is believed to have not formed in the normal way or time, but somewhat later than the formation of Earth, when another planet collided with Earth and formed an orbital ring of debris from which the Moon formed.
Such a planetary would have been much more common if the first few hundred million years of the solar system, before planetary orbits stabilized, than after things settled down. But it could still happen in a later stage of a star system, although it would be very unlikely.
But the collision which formed the Moon would have made the Earth's surface a mass of red hot lava, destroying any life forms which might have appeared before the collision. The first lifeforms that we are descended from must have appeared many millions of years after the collusion, after Earth had cooled.
And after those life forms appeared it took billions of years for the activities of life to produce an oxygen rich, breathable atmosphere for Earth only about six hundred billion years ago.
So if the hypothetical collision producing the hypothetical moon of Proxima Centauri b happened only six hundred million years after the collision which formed Earth's Moon, the planet Proxima Centauri b might not yet have a breathable atmosphere and the moon might have already have escape from orbit around Proxima Centauri b and might already be a menace to Proxima Centauri b, possibly colliding with it in the future.
There is one other case in our solar system where moons are believed to have formed by a collision between larger bodies:
Pluto's moons are hypothesized to have been formed by a collision between Pluto and a similar-sized body, early in the history of the Solar System. The collision released material that consolidated into the moons around Pluto.
Pluto's largest moon, Charon, is not only tidally locked to Pluto, but has already tidally locked Pluto to itself.
The Pluto–Charon system is one of the few in the Solar System whose barycenter lies outside the primary body; the Patroclus–Menoetius system is a smaller example, and the Sun–Jupiter system is the only larger one. The similarity in size of Charon and Pluto has prompted some astronomers to call it a double dwarf planet. The system is also unusual among planetary systems in that each is tidally locked to the other, which means that Pluto and Charon always have the same hemisphere facing each other — a property shared by only one other known system, Eris and Dysnomia. From any position on either body, the other is always at the same position in the sky, or always obscured. This also means that the rotation period of each is equal to the time it takes the entire system to rotate around its barycenter.
Simulation work published in 2005 by Robin Canup suggested that Charon could have been formed by a collision around 4.5 billion years ago, much like Earth and the Moon. In this model, a large Kuiper belt object struck Pluto at high velocity, destroying itself and blasting off much of Pluto's outer mantle, and Charon coalesced from the debris. However, such an impact should result in an icier Charon and rockier Pluto than scientists have found. It is now thought that Pluto and Charon might have been two bodies that collided before going into orbit about each other. The collision would have been violent enough to boil off volatile ices like methane (CH4) but not violent enough to have destroyed either body. The very similar density of Pluto and Charon implies that the parent bodies were not fully differentiated when the impact occurred.
And once both Pluto and Charon were tidally locked, tidal interactions seem to have stopped pushing Charon away from Pluto. Pluto has a radius of about 1,188.3 kilometers and Charon has a radius of about 606.0 kilometers.
The semi-major axis of Charon's orbit around Pluto is about 19,591.4 kilometers. Which means that the average separation between the surfaces of Pluto and Charon should be about 16,608.8 kilometers. And Charon should have been pushed out from Pluto from its formation distance until Pluto became tidally locked to Charon.
So if a hypothetical moon of Proxima Centauri b formed very close to the planet, and was much larger and more massive relative to the planet than the moon is to Earth, is could have tidally locked the planet to it and ceased to be pushed outward from the planet before leaving the region of true stability.
The hypothetical moon is similar in size and mass to the Earth's Moon and orbits the planet at a similar distance. As the planet has more mass than Earth, the moon orbits it in 575.3 hours.
As I have show, it is very improbable for Proxima Centauri b to have such a moon. A moon would probably have to be much larger and closer to Proxima Centauri b to have a long term stable orbit and it would probably have have to have tidally locked Proxima Centauri b to itself.
I don't know if the hypothetical moon could have tidally locked Proxima Centauri b to itself before Proxima Centauri locked Proxima Centauri b to itself. Or rather, I don't know what combinations of mass and initial orbit of the moon would enable it to win the race to tidally lock Proxima Centauri b to it.
But due to the short distance, the planet is tidally locked to its host star, and the same side is permanently facing the star. That, I assume, would constrain what moon phases could be seen from a particular point.
That is inaccurate.
Proxima Centauri b is likely to be tidally locked to the host star, which for a 1:1 orbit would mean that the same side of the planet would always face Proxima Centauri. It is unclear whether or not habitable conditions can arise under such circumstances as a 1:1 tidal lock would lead to an extreme climate with only part of the planet habitable.
However, the planet may not be tidally locked. If the eccentricity of Proxima Centauri b was higher than 0.1-0.06, it would tend to enter a Mercury-like 3:2 resonance[f] or higher-order resonances such as 2:1. Additional planets around Proxima Centauri and interactions[g] with Alpha Centauri could excite higher eccentricies. If the planet isn't symmetrical (triaxial), a capture into a non-tidally locked orbit would be possible even with low eccentricity. A non-locked orbit however would result in tidal heating of the planet's mantle, increasing volcanic activity and potentially shutting down a magnetic field-generating dynamo. The exact dynamics are strongly dependent on the internal structure of the planet and its evolution in response to tidal heating.
Thus at the present time astronomers are uncertain whether Proxima Centauri b is tidally locked in a 1:1 resonance with Proxima Centauri or whether it does rotate with respect to Proxima Centauri and thus has days and nights.
But the question asks about about Proxima Centauri b having a moon. And I have shown that it is almost impossible for Proxima Centauri b to have kept any moon until the present time unless that moon has tidally locked Proxima Centauri b to itself and has stopped being pushed outwards.
Thus Proxima Centauri b and its hypothetical moon should be tidally locked to each other and thus should rotate with respect to Proxima Centauri and experience alternations of night and day on their surfaces.
Since Proxima Centauri b and its hypothetical moon should be tidally locked to each other, the moon should be permanently hidden from about one half of Proxima Centauri b's surface and the moon should permanently visible in the sky of about one half of the surface of Proxima Centauri b. And as seen from the parts of the surface of Proxima Centauri b where the moon is visible, it should show all the phases from new to full that Earth's Moon shows.
The only difference between the visual aspects is that on a one spot on the surface of Proxima Centauri b, the moon would either always be out of sight or else always appear in the same spot in the sky, not appearing to move relative to the surface. And that the period of time for the the moon to go through its cycle of phases would be different than the Moon's period, since it would equal the orbital period of Proxima Centauri b and its moon around each other.