And thus we descend into the complicated post-Alexandrian mess!

When we talk about succession to Alexander, really we're talking about succession to two different things- his successor as King of Macedon, and his successor as King of Asia (i.e Emperor of the now-ex Achaemenid domains). In theory, as with Alexander, these two roles would be united in the same person, but the 'rules' for succession in both cases were not identical.

In the case of the Kingdom of Macedon the succession, in terms of theoretical precedence, was clear; the ruling dynasty was that of the Temenids, who in modern histories are usually referred to as the Argeads (the name was due to their claim of descent from Temenus, a descendant of Herakles). The king would always come from that royal family, and Macedonian kings would designate their heirs (who were usually their eldest son). There was input from other Macedonians- the Assembly of the Macedonians had to approve the choice of successor after the previous king had died. But, in theory, you would have assumed that the male sons of a King would inherit his title, and if not a son then a brother or another close male relative. Things did often not work out that simply in Macedon, however- the royal family was legendarily prone to murdering one another and usurping the throne. Phillip II and Alexander III (i.e the Great) represent one of the few relatively uncontested periods in Macedonian history by this point, and Phillip II had himself effectively usurped the throne from his own nephew in the first place.

Given the usual run of things it would have been assumed that Alexander's firstborn son, Alexander IV, would have been made King of Macedon. However, Alexander's imperial position and possessions complicated matters, as I'll now go into.

Alexander did not claim the territories and title of the Achaemenid Empire as extensions of Macedon, or under the aegis of being King of Macedon. From all accounts we have he was actively claiming his approximation of the Achaemenid King's own title, formulated in Greek sources as 'King of Asia'. What possible legal claim could he have to the title of the Achaemenid monarchs? None, and nor did he claim one. Interestingly many later Persian accounts of Alexander have him descended from the Achaemenid dynasty, giving him a legitimate claim to the title, but this is a later rationalisation and not one used at the time. Instead, his claim on the Achaemenid Empire was framed by Greek historians as being 'spear-won-land', ge doryktetos. It is literally claiming right of conquest by the fact that you conquered it in the first place. It is a formulation that deliberately harkens back to imagery used in the Homeric epics, but here it is given a particular twist- if Alexander was capable of conquering the Achaemenid Empire, he was capable of ruling it. He possessed no dynastic claim of his own, and his legitimacy was not conferred from that. His son, who was born the same year that he died, was born to his wife Roxane. She was herself royalty, but she was part of a royal family from Bactria, merely subordinate client kings under the Achaemenid (and now Temenid) rule; her family had no more dynastic claim to the Achaemenid throne than Alexander himself. However, Alexander did marry at least one relative of Darius III, and it is believed that this is why Roxane had Alexander's other wives murdered after his death- if they claimed to be pregnant and gave birth to a baby regarded as Alexander's child, they would possess a much more apparent dynastic connection to the Achaemenids.

The assumption was that Alexander IV should have inherited both of his father's positions, but there was already a tension- Alexander's claim on the title King of Asia was based on conquest, not on dynastic heritage. But this was immediately made more confusing by the fact that Alexander IV was a newborn child, necessitating a regency. Likewise he also began far away from Macedon itself, in Babylon. This is when you get the earliest part of the Succession Wars, in which regent after regent was killed or assassinated, and in which various parts of the Temenid dynasty went to war with other parts, allied again, then went to war again. Alexander's favoured brother, Phillip III, was effectively declared king in Macedon, but he was either mentally disabled or had received brain damage at some point and was thus manipulated by others. He was eventually killed by Alexander's own mother, Olympias. The end result was that Alexander IV did end up in Macedon, and the peace that ended the initial Civil Wars in 311 BCE was predicated upon him succeeding to the Macedonian throne once Cassander (one of Alexander's generals who himself possessed no dynastic claim on the throne) died. However, once demands were made for Alexander IV to be crowned Cassander had him assassinated.

By the time Alexander IV had died almost the entire rest of the Temenid dynasty had been killed. Olympias was dead, Phillip III and his wife Eurydice (who was herself a Temenid) was dead. Any notion of dynastic legitimacy was gone. The only surviving direct descent of Phillip II we hear of from this point is Cleopatra, who was the father of Alexander of Molossia, a king of Epirus. This made Alexander of Molossia Alexander's uncle, but he died even before Alexander did on campaign in Italy. His own son, Neoptolemus II, was therefore closely related to Alexander. However, he was murdered by the famous Pyrrhus after becoming his co-ruler. Pyrrhus himself was related to Alexander via his family connections to Olympias, but was not personally part of the Temenid dynasty. Some ancient sources suggest Ptolemy was a bastard son of Phillip II, but so far as I know none of them state it outright and instead simply say that this was a common rumour. It's quite likely that it was a rumour either started or encouraged by Ptolemy himself, as connections to Alexander were the currency of legitimacy among the successors to Alexander.

So, we're without any dynastic successors to point out- Macedon came under the rule of new dynasties unrelated to Alexander's Temenid dynasty, and ditto with his other territories. But what of the principle of spear-won-land, what of ge doryktetos? In that respect all of the successors who successfully forged dynasties and claimed territories were equally legitimate successors. Many of them had to fight very dirty to get to that point, and some of those dynasties barely lasted a generation before being squashed by their rivals, but nobody said that winning land by the spear involved being nice- Alexander himself killed a huge portion of his extended family upon the assassination of Phillip II. However, in terms of being a direct successor to Alexander I would suspect maybe the Seleucids have the edge. It was they who grabbed hold of the titles associated with being 'King of Asia', and who effectively inherited the Achaemenid Empire's territories in Asia, along with most of their own titles. By the principle of ge doryktetos the sheer fact that they gained these titles in the first places made them the legitimate successors. Similar logic applied to some of the other titles they gained, such as being Kings of Babylon- the Kings of Babylon were assumed to be chosen by Marduk himself, and Marduk's usual way of having his will made manifest was by somebody basically beating all of his opponents to claiming the kingship. However, in terms of the basis of their claims ultimately none of the successor kingdoms had better claims than the others. This is why they competed so fiercely over their connections to Alexander, over dictating the historiography of Alexander's reign, and over imagery relating to Alexander in the first place.