Well, that's the end of the Persian expedition right there. The succession is a complete mess, since Alexander killed Amyntas exactly because he could theoretically be a competitor. Amyntas had been married to Alexander's eldest half-sister Kynane, who is as such available for marriage to cement the claim of any would-be claimant. Alexander's full sister, and arguably with a better claim (having the backing of Olympias, no doubt) was Kleopatra-- also widowed, also available as a bride for any would-be claimant. Alexander's half-brother Philipos Arrhidaios was not considered a threat bt Alexander (even though his claim was at least as strong as that of Amyntas), and it's assumed he was epileptic to an extent that rendered him unable to be a credible threat. (It is sometimes assumed he was a half-wit, too, but this is pure conjecture.)
In any event, the army abandons the campaign and returns across the Hellespont. This may be a good moment for Ptolemaios to reveal (or "reveal") that those stories about him being an illegitimate son of Philippos II are true (or "true"), and put forward his claim. He was popular, gifted, and while the would be rivals, he had always been loyal to Alexander-- which he could use to his advantage, claiming that Alexander knew of his true parentage, but had accepted him as a brother etc. etc.
One suspects that if Ptolemaios makes a strong case and wants to marry Kleopatra, that would go through and bolster his claim. (Don't worry about the fact that he's -- at least supposedly -- her half-brother; that never bothered the Greeks, who only considered it incest if people are full siblings. Macedonians certainly never cared about any of this; Kleopatra's first husband was her uncle.)
Anyway, we may expect there to be several nominal claimants. Depending on how successful they all are, there might be short struggle followed by a clear winner (my money, if you hadn't guessed, is on Ptolemaios), or there might be a long civil war. There's the possibility of the "old guard" of Philippos II (so Antipatros, Parmenion, Krateiros, Kleitos etc.) attempts to side-track Alexander's young Companions, in which they would draw their sons and other relatives into their own camp. It's also possible these older generals fall out with each other, unable to decide which of them is their candidate-- leading to more senseless bloodshed. If they do unite as one faction with their own 'candidate', I suspect Antipatros would be their claimant. It is also possible that a faction props up Arrhidaios as a puppet king. The old guard is most likely to try that, in which case they'd form a cabal with a lot of legitimacy bwhind their claim.
I could see, just for instance, Ptolemaios marrying Kleopatra and having the backing of most of Alexander's tight-knit group of Companions, as well as Olympias. Against them would be the cabal of the Old Guard, propping up Arrhidaios, who is married off to Kynane with the uderstanding that this union won't be expected to produce children, and that Kynane's daughter (by Amyntas) will at some future point be married to an appropriate heir-- meaning a son from one of the cabal members... probably Kassandros, son of Antipatros. Well. ENJOY YOUR CIVIL WAR. Greece will no doubt exploit this to try and regain independence. Depending on how messy and how dragged-out the Macedonian Civil War gets, the poleis may just succeed. In which case you may expect Demosthenes to lead the impromptu conga line at the resulting party in Athens.
In the meantime, the Akhaimenid Empire would bounce back from its recent troubles. Keep in mind, Alexander invaded in the wake of a civil war / succession struggle, and really got to exploit the mess that the empire hadn't yet recovered from. Had he been ten years later, it would have been a different story. Had he been anyone else, the same goes. The myth of Persian decadence and weakness is total bullshit, and Alexander was both uniquely talented and happened to strike at a highly convenient time (highly inconvenient for Persia, that is). So you may expect the Akhaimenids to thrive, and the resounding defeat of the invasion would even be a major boon to the dynasty's credibility and esteem.