Alexander III was an explosively violent, alcoholic, paranoid, obsessive, narcissistic megalomaniac. He was also sentimental and grandly romantic of course ( not to mention an undeniably charismatic, brilliant general ). But a happy family court equivalent to Attalus I’s in Pergamun seems highly unlikely. It may well be that an adult Alexander “IV” would be at nearly as much threat from from his volatile, suspicious father as he ended up being from Cassander. Oh, no doubt Alexander III would have wailed and tore his clothes after stabbing his son in the liver for arguing with him over wine, but small consolation that would have been ;).
Also Alexander was obviously a fertile-enough fellow. Had he lived longer we might have been looking at multiple competeing princes all vying for control at his death.
It is likely Alexander would have turned his restless ambition west towards places like Magna Grecia, the young Roman state and Carthage. The outcomes of any such campaigns are of course beyond speculative, but I guess I wouldn’t be the one to bet against Alex. But I disagree that his state was stable at his death - it was a ramshackle, incompletely conquered edifice. Even at his death sizeable chunks of Asia Minor, bypassed by his rapid movement, were unconquered and Greece was about to explode into revolt. These challenges would likely have continued, even as he marched westward. While he may well have contained them while he lived, decentralization was probably inevitable, with likely an Achaemenid-style satrapy system which would have been ripe for implosion into secessionist states as happened during the Wars of the Diadochi.
I guess two big speculations I could throw out there that might substantially change history is:
1.) Following an Argead conquest, a Magna Grecia-based ( because it surely wouldn’t center in Latium ) Diadochid state that encompassed Italy and pre-empted the rise of Rome. The possibilities here for change are conceivably quite profound. They might be speaking a bastardized Greek dialect in France today, instead of a Latinate tongue.
2.) An Argead heir, son of an Achaemenid princess, raised in Babylon, forming a far more cohesive and unified Greco-Persian empire than that raised by the Seleucids ( who were as throughly bigoted and exclusionary against non-Greeks as most Macedonians, Peucestes excluded ). That could potentially be a true powerhouse.