According to Peter Brown's Augustine of Hippo and The World of Late Antiquity, what were some important religious and philosophical movements and...

The main religious developments of the fourth century stemmed from Constantine's conversion to Christianity. Starting with the Edict of Milan in 313 which legalized Christianity, there was a gradual shift from the traditional gods being linked with imperial power to Christianity becoming the official religion of the Roman Empire. 

When Christianity had been an underground religion promulgated by a persecuted minority, having a clear and authoritative hierarchy and uniform doctrine had been less important than...

The main religious developments of the fourth century stemmed from Constantine's conversion to Christianity. Starting with the Edict of Milan in 313 which legalized Christianity, there was a gradual shift from the traditional gods being linked with imperial power to Christianity becoming the official religion of the Roman Empire. 

When Christianity had been an underground religion promulgated by a persecuted minority, having a clear and authoritative hierarchy and uniform doctrine had been less important than simple survival, although even as early as the late first century we find many internal doctrinal differences within Christianity. As an official imperial religion, Christianity needed to be strongly centralized and uniform. The Council of Nicaea marks the beginning  of collaboration between the episcopate and the Emperor to make Christianity an imperial religion, functioning as part of the Roman state. Stamping out of heresy and suppressing dissent became a priority.

The fourth century was also a period of the blossoming of monasticism, especially in the Greek east. Also, it was a period in which Augustine and subsequent thinkers argued that classical learning could be assimilated into Christian thought in the manner of the Egyptian gold the Jews had appropriated during the Exodus. 

In pagan philosophy, the fourth century was a period of syncretism, with Platonist philosophy, mystery religions, and Hellenic culture blending to create a more monolithic opposition of traditional paideia to the new Christian religion. Platonism became more heavily Pythagoreanized during this period, moving from the skepticism of the middle Platonists to a more mystical form of neoplatonism, characterized by the work of Plotinus, Iamblichus, and Proclus. This was also a major period for commentaries, both on the Platonic dialogues and Aristotle, whose work was assimilated to Platonism by Porphyry and subsequent Platonists.