In terms of an heir, he seems to have been loose in his definition, and for a while at least his son Alexander IV was 'it' on paper - until the boy and his mother Roxanne were murdered on the orders of Cassander at Amphipolis, and secretly buried.
His plans for expansion and the invasion of the Punic west Mediterranean, as described in Diodorus Siculus (18.4.1-6) were quickly abandoned, as were the proposed population transfers.
It happened that Craterus, who was one of the most prominent men, had previously been sent away by Alexander to Cilicia with those men who had been discharged from the army, ten thousand in number. At the same time he had received written instructions which the king had given him for execution; nevertheless, after the death of Alexander, it seemed best to the successors not to carry out these plans.But what about the architectural plans also described in Diodorus?
For when Perdiccas found in the memoranda of the king orders for the completion of the pyre of Hephaestion, which required a great deal of money, and also for the other designs of Alexander, which were many and great and called for an unprecedented outlay, he decided that it was inexpedient to carry them out.Earlier Diodorus (17.114‑115) had already covered the pyre and funerary procession of Hephaestion, which took place whilst Alexander was still living, so presumably this is a confusion. I'd suggest that Perdiccas refused state funding to the final parts of the funerary obsequies decreed by Alexander for Hephaestion: a magnificent tomb and the setting up of cults to him as a hero (for which see Plutarch, Alexander, 72).
Andrew Stewart in Faces of Power: Alexander's Image and Hellenistic Politics (Berkeley 1993; p. 213 n. 69 and chapter 3) points out that this late 4th century BC relief from Pella (now Thessalonike Museum 1084; see most recently Olga Palagia in Brill's Companion to Ancient Macedon, 2011,
p.491, available as a PDF here) is the only certain evidence for a cult of the Hero Hephaestion in Greece. It is dated to soon after his death, and nothing else excavated so far has suggested that there was any cult that survived the death of Alexander the Great outside Egypt. There is a head in the Getty said to be of Hephaestion, said to be from a monument from Messene, said to be ... but alas it has no provenance other than the art market, and I doubt even its authenticity.
Continuing on the text in Diodorus Siculus (18.4.1-6), Perdiccas:
But that he might not appear to be arbitrarily detracting anything from the glory of Alexander, he laid these matters before the common assembly of the Macedonians for consideration.
The following were the largest and most remarkable items of the memoranda. It was proposed to build a thousand warships, larger than triremes, in Phoenicia, Syria, Cilicia, and Cyprus for the campaign against the Carthaginians and the others who live along the coast of Libya and Iberia and the adjoining coastal region as far as Sicily; to make a road along the coast of Libya as far as the Pillars of Heracles and, as needed by so great an expedition, to construct ports and shipyards at suitable places;They voted for none of the above.
to erect six most costly temples, each at an expense of fifteen hundred talents; and, finally, to establish cities and to transplant populations from Asia to Europe and in the opposite direction from Europe to Asia, in order to bring the largest continents to common unity and to friendly kinship by means of intermarriages and family ties.
The temples mentioned above were to be built at Delos, Delphi, and Dodona, and in Macedonia a temple to Zeus at Dion, to Artemis Tauropolus at Amphipolis, and to Athena at Cyrnus. Likewise at Ilion in honour of this goddess there was to be built a temple that could never be surpassed by any other. A tomb for his father Philip was to be constructed to match the greatest of the pyramids of Egypt, buildings which some persons count among the seven greatest works of man.
When these memoranda had been read, the Macedonians, although they applauded the name of Alexander, nevertheless saw that the projects were extravagant and impracticable and decided to carry out none of those that have been mentioned.Again these projected were voted down.
If Tomb II at Vergina is the final burial place of Philip II - and the arguments are more convincing that those that it was the tomb of his son Philip III Arrhidaeus - and considering there are no literary records of it, we can safely assume that Alexander the Great's successors ignored the wish that "A tomb for his father Philip was to be constructed to match the greatest of the pyramids of Egypt"
But what about the temples? They may not have been authorised and funded under the regency of Perdiccas, but archaeological evidence suggests that at least some of them were built by the Diadochi.
Dion was sacked during the reign of Philip V, and the temple of Zeus destroyed (Polybius 4,62.2) so it is difficult to know whether the later temple replaced an early Hellenistic one or the one that had stood in Alexander's reign.
Alexander's heirs did not build the temple of Apollo at Delphi, and I cannot think of a suitably magnificent temple that could be ascribed to them. Ditto Delos and Dodona, as well as Cyrnus, but those might more reflect lacunae in my knowledge.
As to the temple "to Artemis Tauropolus at Amphipolis" ... the town is still being excavated - although some of you might have noticed the archaeologists are a little busy with a tomb outside Amphipolis - but there is good evidence of a Hellenistic cult of Artemis Tauropolus there. For example the 2nd century BC Roman period coins of the town, which continued the Antigonid practice of having a shielded head, but with the head of Artemis Tauropolus and sometimes a bull on the reverse (for more info see S. Kremydi in Brill's Companion to Ancient Macedon, 2011, p.175; on Google books here).
Sources of photos: the Tetradrachm; the bronze with Artemis Tauropolos riding bull is one several shown at Wildwinds here.
There are also literary sources attesting the cult, and meetings held at it, as is clear in Wandering Images: From Taurian (and Chersonesean) Parthenos to (Artemis) Tauropolos and (Artemis) Persike, in xxx, Aarhus 2003, available as a PDF here, and from whom this map comes:
When it comes to Troy and "Likewise at Ilion in honour of this goddess there was to be built a temple that could never be surpassed by any other" we are on firmer ground as there are the remains of an early Hellenistic temple to Athena; it may not have surpassed all other temples, but it was lavishly decorated, and had sculpted metopes all around it. Most imitated the Parthenon in terms of its themes - including, ironically, one side showing the Iliupersis where the 'civilised' Greeks defeated the 'barbarian' Trojans ... - but one metope, now in Berlin, showed Helios with rather leonine locks:
The problem is that the temple was found in pieces, partly by Frank Calvert, the discoverer of Troy, and partly by Schliemann the showman who ripped through the site to try to find the Homeric city.
I wrote extensively about the sculptures and the temple of Athena at Ilion in my PhD, and way back in the last century we were convinced that it had been built by Lysimachus (died 281 BC), as recorded by Strabo (13.1.26):
It is said that the city of the present Ilians was for a time a mere village, having its temple of Athena, a small and cheap temple, but that when Alexander went up there after his victory at the Granicus River he adorned the temple with votive offerings, gave the village the title of city, and ordered those in charge to improve it with buildings, and that he adjudged free and exempt from tribute; and that later, after the overthrow of the Persians, he sent down a kindly letter to the place, promising to make a great city of it, and to build a magnificent sanctuary, and to proclaim sacred games. But after his death Lysimachus devoted special attention to the city, and built a temple there and surrounded the city with a wall about forty stadia in circuit, and also incorporated into it the surrounding cities, which were now old and in bad plight.The German publishers had taken Augustus' claim in the inscription to have rebuilt the temple literally, and dated it to his period. Most scholars had argued that they were wrong, and that it was an early Hellenistic construction - so that the Roman claim of capture and destruction was a literary topos. There were also suggestions that Antigonus I Monophthalmus had initiated the work before he died in 301 BC, and that it was completed by Lysimachus. These two, rival and successor kings of Macedon were thus completing Alexander's wish in the territory they held, and by doing so staking their claim to succeed him.
But ... Charles Brian Rose, who's in charge at Troy, has re-examined the old excavations and now feels that the temple was built on foundations who pottery comes a generation after them. He amends the Strabo passage to refer to Lysimachus building a now lost temple at Alexandria Troas, and prefers to see Antiochus III Hierax as the builder of the temple of Athena at Ilion, possibly completed by the Attalids. The abstract of his article, The Temple of Athena at Ilion, Studia Troica 13, 2003, pp. 27ff, can be found here; and in The Archaeology of Greek and Roman Troy, Cambridge 2013, pp. 168ff (on Google books here).
I've included this as Rose is the great authority on Troy. My view is that Hierax was too busy fighting against his brother for the Seleucid throne to worry about a promise to rebuild an obscure temple far outside his homeland. Because of the way the site was bulldozed by Schliemann, I am not entirely convinced by the old evidence, but ... I also admit I spent so much time arguing against the Roman and for the Lysimachus date in my thesis that I might not be objective, which makes me sceptical about the redating of very old digs.
Anyway, the point is that at least at one location - Troy, where Alexander's beloved Achilles was buried - Alexander's wishes were fulfilled by his successors. And the Strabo text suggests that this point was made by the builder, and remembered and repeated by the later inhabitants of the city.
So were Alexander's wishes fulfilled? The answer seems to be 'not really ... except perhaps sometimes by successors trying to stake a claim to being his heir'.
See also Arachne here for another fragment in Istanbul; a fragment in Cannakale; Athena slaying a Giant, and another piece in Berlin.
The Calvert fragments are lost, but I found photos of them in a box labelled "Calvert" at the DAI Athens photo library.
(I'll try to find more photos, but am still recovering from being flooded ... just be grateful I didn't cut and paste dozens of pages of my dissertation to try to back up my point ...)