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Alhazredian was the root style from which all modern systems stem, the Indo-European analogue, if we are to compare systems to real life languages. It was clumsy, clunky and badly put together, with no unifying structure or standardised basis for spells. Components of old spells were welded together to get new effects with no streamlining or optimisation, and the priest-kings developed their own from scratch, customising them and leaving no notes as to their methods. A common method of designing a new spell for a commoner was to take several existing spells that did something similar to the desired effect, remove the parts of each spell-structure that obviously caused tangential or unwanted effects, and then graft the remainders together in various different ways until the desired effect was achieved. This effectively produced layer upon layer of reused modules grafted onto one another in hopelessly tangled spaghetti code, and is the leading reason that surviving or uncovered spells from that era are almost completely unintelligible. They work, but they are generally slow to cast, requiring chants lasting anywhere from several seconds to several minutes to activate, and highly inefficient, wasting a lot of mana in redundant cycles. Additionally, exactly how they do what they do - the mechanisms by which they cause the outcome they produce - are essentially impossible to determine due to the complexity and incoherent chaos of the spell structure.
Such spells were also costly. The median Alhazredian would have been rather pathetic compared to the median mage in dimensional space, knowing at most a few spells rote-memorised for their role in life, and would be unable to cast many in succession. The Alhazredian anomalies come from the priest-kings and the priest-savants, limited numbers of freakishly powerful mages. While their occurrence was no more common than normal, the Alhazredians had effective life-extension treatment, and thus a large fraction of the most powerful mages lived hundreds or even thousands of years, so there were far more of them around at a time. They also had lots of children, and used methods such as Familiar-Boosting and high energy density artifacts such as the Jewel Seeds to draw from to make up for the prohibitive costs of their spells.
The personal spells of the priest-kings themselves were not produced in this slap-hazard, clumsy manner. Unfortunately, this makes them no more comprehensible. A priest-queen would design her own spells individually, from first principles, each and every time. As a result, each spell is a custom-built structure designed from a mix of pre-existing functions and its creator’s own idiosyncratic, personal way of approaching things, often using radically different methods to achieve the same effects as another priest-savant’s spells. It was common for them to work in booby-traps, so that attempting to remove elements from the spell and then casting it to see how it differed from the complete version (and thus understand that element’s function) was liable to turn it into a suicide technique or cause it to backfire lethally. Defences like these prevented the spells being stolen, reverse-engineered and then used against the priest-savant who had created them, as did the remote-crippling codes which only they knew about in the versions they taught to their subordinates. As such, the study of recovered and surviving priest-king-designed spells has met with little more success than commoner ones.
These advanced spells took precious time to cast, and so the Alhazredians came up with ways to protect themselves while doing so. One of these was of course the summoning lineages; warrior-priests and priestesses who were the favoured champions of the priest-kings, engineered for their roles as warriors and sacred prostitutes alike. Others were shock troops, taught a group of related combat spells with similar structures and set to guard their masters as they chanted the spell-hymns to unleash their devastating magic - and it is from these, expanding on the already-similar underpinnings of their arsenals to come up with standardised systems of spellcrafting, that the first magical styles developed after the fall of Alhazred.
Despite its failings, the magic of the Alhazredians excelled in one thing, and that was its scope. Certain of its effects, such as the Familiar creation spell, have never been fully replicated, and many of them accomplish things which modern magical systems are still incapable of. While the commoner magic is inferior in almost every respect to modern styles, the spells of the priest-kings could, given five or six minutes to chant, unleash a mana-charged stormcloud onto a region which rips a cubic centimetre around each raindrop into Imaginary Space when they hit the ground, rendering the storm as a whole capable of reducing entire cities to nothing in the space of mere minutes.
Visually, Alhazredian had no unified structure, and therefore no geometric spell-sigils. The light that appeared around them as they cast took the form of scattered cuneiform clumped into rough masses and linked with lines that often passed over one another or through other clumps. Some proto-standardised styles did exist in the form of one man or woman's approach to doing things, but such things were kept, at most, to their circle, as they had no wish to share the secrets of squeezing a little more efficiency out of their spells with other priest-kings. Even these were more along the lines of personal "function libraries", wherein the proto-style comes more from the fact that the priest-queen was using the same names for each variable and thus using similar structures for her spells than anything else, though this was still enough for the cuneiform to take somewhat predictable patterns as a recognisable signature.