One argument for independence is their differing accounts of the conflict between Peter and Simon Magus.
Pseudo-Hegesippus dates this in the time of Nero.
http://www.tertullian.org/fathers/heges ... _book3.htm This agrees with various apocryphal Acts and derives from the old tradition that Peter came to Rome in the reign of Nero.At that time Peter and Paul were in Rome, teachers of the Christians, distinguished in works, brilliant in administration, who by virtue of their works had made Nero an adversary who had been captured by the enticements of the Magian Simon who had won over his mind to himself. To whom he promised with deadly arts the aid of victory, the subjections of peoples, longevity of life, the protection of safety, [p. 184] and he believed who did not know to examine the meaning of things. Finally he held the highest place of friendship with him, seeing that he was even considered the chief of his security and the guardian of his life. But when Peter uncovered his falsities and faults, and showed appearances to deceive him of things, and not to produce anything real or true, he was consumed with grief and considered an object of and worthy of mockery. And although in other parts of the world he had experienced the power of Peter, however preceding (Peter) to Rome he dared to boast, that he had restored the dead to life. There had died at that time at Rome a young noble relative of Caesar to the sorrow of everyone. Many suggested it should be tried whether he could be restored to life. Peter was considered the most renowned in these tasks, but among the gentiles no trust was accorded to achievements of this sort. Grief demanded a remedy, recourse was had to Peter. There were even those who thought Simon should be summoned. Both were at hand. Peter says to Simon, who was boasting about his ability, he would give (him) the first chances as if he were able to revivify the dead man. If he did not revivify (him), he would not be absent when Christ should carry succor to the dead man, at which time he would be able to arise. Simon, who thought his arts would be especially strong in the city of gentiles, proposed the condition that if he himself should revivify the dead man, Peter should be killed, who had proposed great authority, for it was named thus, by calling forth insults, but if indeed Peter should have superior power, he should in like manner make a claim against Simon. Peter assented, Simon made the attempt. He approached the bier of the dead man, he began to fix a spell and to murmur fearful incantations. He who was dead was seen to shake his head. a great clamor of the gentiles because he was now living, because he was speaking with Simon. Anger and displeasure against Peter because he had dared to compare himself to such great ability. Then the blessed apostle [p. 185] demanded silence and says: 'if the dead man is alive, let him speak; if he has been revivified, let him stand up, walk, converse.' That to be an illusion, not reality that he seems to have moved his head. Finally he says "let Simon be separated from the funeral-bed,' and then indeed it will not be a pretence. Simon is led away from the bed, he who was dead remains without the appearance of any motion. Peter stood farther away and intent within himself for a short time on his words he says with a loud voice: 'Young man, stand up: the lord Jesus makes you well.' Immediately the young man rose up and spoke and walked and took food and Peter gave him to his mother. Who when he was asked that he should not depart from him said: 'let him not depart from him who made him to rise up, whose servants we are. Be untroubled, mother, about your son, do not fear, he has his protector.' And when the people rose up against Simon that he should be stoned, Peter says: 'It is enough for his punishment that he knows his arts to have no strength. Let him live and even though unwilling see the kingdom of Christ to grow.' The Magian turned away from the glory of the apostle. He collected himself and calls forth all the power of his enchantments, he assembles the people, he says himself offended by the Galilaeans and to be about to leave the city which he was wont to guard. He set a date, he promises that he will fly, where he would be borne on celestial thrones, to whom when he should wish the heavens would open. On the appointed day he ascended the Capitoline hill and throwing himself from the rock he began to fly. The people marvelled and many worshipped saying the ability was god's, it was not a man, who should fly with a body, Christ to have done nothing similar. Then Peter standing in their midst says: 'Jesus Lord, show him his arts to be empty, lest by this display this people who are about to believe should be deceived. Let him fall, lord, in such a way however, that living he will recall himself able to do nothing.' And immediately on the words of Peter with a tangling of the wings [p. 186] which he had put on he fell to the ground, nor was he killed, but with a broken and lamed leg he withdrew to Aricia and died there. This having been learned Nero grieving himself deceived and abandoned by the fate of such a great friend, a man useful and necessary to the state having been taken away from him, indignant began to seek causes for which he should kill Peter. And already the time was at hand when the blessed apostles Peter and Paul would be called. Finally the order having been given that they should be seized Peter was asked that he should take himself elsewhere. He kept resisting saying he would never do it, as if terrified by the fear of death he should yield, it was good to suffer for Christ, who in behalf of all offered himself to death, not that death but immortality to come, how unworthy that he himself should flee the suffering of his body. Who by his teachings brought many to offer themselves as sacrifices for Christ, to be owed to himself according to the voice of the lord, that he himself also should give glory and honor to Christ in his suffering. This and other things Peter hid, but the common people sought with tears, that he should not abandon himself, and that he should abandon the wavering between the commotions of the gentiles. Overcome by the lamenting Peter conceded, he promised himself to be about to leave the city. The next night with brothers taken leave of and solemn speech he began to depart alone. When it was come to the gate, he saw Christ to be coming to meet him and adoring him he said: 'lord, for what purpose do you come?' Christ says to him: 'I come again to be crucified.' Peter understood it was said about his own suffering, because in him Christ would be seen about to suffer who suffered for every one, not certainly by pain of body but by a certain fellow-suffering of sympathy and celebration of glory. And turning around he returned into the city and was captured by pursuers. Condemned to the cross he demanded that he be fixed to the cross feet uppermost, because he was unworthy that he should be fixed to the cross in the same manner, [p. 187] as the son of god had suffered. Which was obtained either because it was owed as Christ had foretold, or because his persecutor not unwilling granted the augmentation of the punishment, and he himself and Paul one on the cross and the other by the sword were killed.
Eusebius however on the basis of Justin's claim that Simon Magus came to Rome in the reign of Claudius dates the conflict in Claudius' time.
The disagreement may indicate that pseudo-Hegesippus did not have access to the works of Eusebius.1. The evil power, who hates all that is good and plots against the salvation of men, constituted Simon at that time the father and author of such wickedness, as if to make him a mighty antagonist of the great, inspired apostles of our Saviour.
2. For that divine and celestial grace which co-operates with its ministers, by their appearance and presence, quickly extinguished the kindled flame of evil, and humbled and cast down through them every high thing that exalted itself against the knowledge of God.
3. Wherefore neither the conspiracy of Simon nor that of any of the others who arose at that period could accomplish anything in those apostolic times. For everything was conquered and subdued by the splendors of the truth and by the divine word itself which had but lately begun to shine from heaven upon men, and which was then flourishing upon earth, and dwelling in the apostles themselves.
4. Immediately the above-mentioned impostor was smitten in the eyes of his mind by a divine and miraculous flash, and after the evil deeds done by him had been first detected by the apostle Peter in Judea, he fled and made a great journey across the sea from the East to the West, thinking that only thus could he live according to his mind.
5. And coming to the city of Rome, by the mighty co-operation of that power which was lying in wait there, he was in a short time so successful in his undertaking that those who dwelt there honored him as a god by the erection of a statue.
6. But this did not last long. For immediately, during the reign of Claudius, the all-good and gracious Providence, which watches over all things, led Peter, that strongest and greatest of the apostles, and the one who on account of his virtue was the speaker for all the others, to Rome against this great corrupter of life. Clad in divine armor like a noble commander of God, He carried the costly merchandise of the light of the understanding from the East to those who dwelt in the West, proclaiming the light itself, and the word which brings salvation to souls, and preaching the kingdom of heaven.
Andrew Criddle
