Matthew 27:3-8 Then Judas, which had betrayed him, when he saw that he was condemned, repented himself, and brought again the thirty pieces of silver to the chief priests and elders, Saying, I have sinned in that I have betrayed the innocent blood. And they said, What is that to us? see thou to that. And he cast down the pieces of silver in the temple, and departed, and went and hanged himself. And the chief priests took the silver pieces, and said, It is not lawful for to put them into the treasury, because it is the price of blood. And they took counsel, and bought with them the potter's field, to bury strangers in. Wherefore that field was called, The field of blood, unto this day.
Acts 1:18-19 Now this man purchased a field with the reward of iniquity; and falling headlong, he burst asunder in the midst, and all his bowels gushed out. And it was known unto all the dwellers at Jerusalem; insomuch as that field is called in their proper tongue, Aceldama, that is to say, The field of blood.
Here are a series of problems claimed of these two accounts.
Matthew has Judas hanging himself, while Acts says he fell over and busted his guts open. So which is correct?
Here are the explanations to be considered:
The standard explanation given by harmonists is that Judas hanged himself, and then his body fell and broke open.
This has some promise: Judas hanged himself on Passover and before a Sabbath, and no Jew was going to touch the hanging corpse (touching a dead body caused defilement; it would have been work to take it down on the Sabbath; added to that, death by hanging was especially a disgrace; and hoisting a dead body isn't an attractive vocation if it isn't on your property), so it is safe to assume that Judas hanged himself and that the branch or rope eventually broke.
Polhill in his Acts commentary [92n] notes that the phrase translated "becoming headlong" (prenes genomenos -- translated as "falling headlong" in the KJV, but literally being "becoming headlong" as shown in Green's Interlinear translation, 366) is a mere transcription error away from being "becoming swollen" (presthes genomenos). The latter may well be what was originally written, and as such might describe Judas' body swelling up after hanging for a while. This reading is found in later Syriac, Georgian and Armenian mss., though perhaps as an attempt at textual criticism of the sort we are doing.
Skeptics do regularly scoff at the suggestion that such a combination of events could happen, and yet be reported differently, but Eddy and Boyd in The Jesus Legend have discovered an almost exactly analogous case involving the lynching of two brothers in 1881.  Two different witness accounts indicated that the brothers were hanged from two different places: a railroad crossing, and a pine tree. Historians would have concluded that there was a contradiction until researchers found photographs proving that both accounts were correct: The brothers had been hanged in both locations, having been apparently first hanged from the crossing, and then later taken down and hanged on the pine tree.
Taken together I still consider the "hanging body/rope broke" solution possible -- but now find something else even more likely. But first let's look at another answer:
Matthew does not even describe Judas' death at all. Here is how one site puts it:
The Greek word translated "hanged himself" is the word apanchomai which is used in Greek literature to mean choking or squeezing one's self as with great emotion or grief. In English we have a similar expression when we say that someone is "all choked up." We do not mean that they have died. We mean that they are overcome with emotion. Judas cast down the pieces of silver in the temple and left doubling himself over with grief.
A check of the lexicons shows that such a meaning is indeed possible, but I found only one actual example listed -- the vast majority of the meanings given were for a physical hanging; there was only one example of a figurative meaning as described. So I would say that this is a possible solution, but not likely.
However, I would now opt for the idea that this is an example of Matthew's creative use of an OT "type". This would combine the idea that Matthew is not actually describing Judas' death, with Matthew's use of the OT texts as typologies.
Audrey Conrad, in "The Fate of Judas" (Toronto Journal of Theology  1992), notes that Matthew's unique words "departed" and "hanged himself" are found in combination in another place in the LXX:
2 Samuel 17:23 And when Ahithophel saw that his counsel was not followed, he saddled his ass, and arose, and gat him home to his house, to his city, and put his household in order, and hanged himself, and died, and was buried in the sepulchre of his father.
Conrad notes that rabbinic interpretation of Ps. 41:9 ("Yea, mine own familiar friend, in whom I trusted, which did eat of my bread, hath lifted up his heel against me.") thought that Ahithophel was the traitor David was describing -- and of course this same verse was applied by Jesus to Judas (John 13:18).
Conrad still thinks there are not enough parallels (!) but we would maintain that the parallels are sufficient, and that Matthew is indeed alluding to the traitor Ahithophel in this passage, and is therefore NOT telling us that Judas indeed hanged himself, but that Judas fulfilled the "type" of Ahithophel by being a traitor who responded with grief and then died. Matthew is thereby making no statement at all about Judas' mode of death, and Luke's "swelling up" stands alone as a specific description of what happened.It makes no sense for the author to tell us that Judas' guts burst without telling us why it happened. Spilling out of guts because of swelling is such a rare event that surely if Luke believed that this extraordinary thing actually happened to Judas, he would have made certain to provide the extraordinary explanation for its occurrence.
This "surely" is the objection of a low-context modern demanding full explanations for every unusual event, but when it comes down to it, neither Luke nor any person could have been able to "provide the explanation" without knowing why it happened. Unless Luke or some other physician had access to Judas (not likely) they could not so much as mount a guess as to "why". (See link 1 below -- secular historians have no problem with similar ideas.)
Matthew says the priests bought the field, but Acts says that Judas did. So who did it?
The alternate site opts for this explanation:
The chief priests did not want to put the money paid for the betrayal of Jesus back into the temple treasury as it was "blood money." So they bought an "agros:" a field to bury strangers in. Because blood money was used to purchase the field it was called "the field [agros] of blood." This is different than the property [chorion] that Judas purchased himself referred to in Acts Chapter 1.
The problem here is that both Acts and Matthew connect the purchase specifically with Judas' act of treachery. Thus I cannot accept this solution. However, it does lead into our own answer. There are a few factors here -- one linguistic, the others sociological.
The word used by Matthew for "bought" is agorazo -- a general term meaning, "to go to market." It means to purchase, but also to redeem. It is a verb that refers to the transaction of business. Note how Luke uses it in opposition to another word:
Luke 22:36 Then said he unto them, But now, he that hath a purse, let him take it, and likewise his scrip: and he that hath no sword, let him sell (poleo) his garment, and buy (agorazo) one.
Poleo can mean "sell" but it's primary meaning has to do with trading and bartering. Therefore the translation of "buy" (and "sell") is made according to context.
How does this mean anything with regard to Judas?
First, note the word Luke uses. It is ktaomai, which means to "get, acquire, obtain, possess, provide, purchase." This word has the connotations of ownership that agorazo does not. Matthew says that the priests transacted business for the obtaining of the field, but they did not thereby have possession of the field. The money they used was Judas' and the field was bought in his name; the field was technically and legally his. (A reader notes that this makes sense because Levites were technically not allowed to own property, so they had to make someone else owner of the field.)
And that leads to another question no one has yet raised, but which I will:
It seems too much of a coincidence, that the priests managed to buy the exact same field that Judas died in.
Not at all. Once Judas died in the field, the land became defiled by his corpse. Hence it would become perfectly suited to become a full-time cemetery. In this ancient collectivist society, the gossip would readily get around as to where and how Judas died and it would not be a burden for the decision to be made to purchase the field in Judas' name (see below) to turn into a cemetery.
If Judas threw the money away, it wasn't his anymore, it belonged to the priests.
This is where our social factor comes into play. Note that the money cannot be put in the treasury -- it cannot be made to belong to the temple again -- because it is blood money. Keener observes in his Matthean commentary [657-8]:
Ancient Eastern peoples regarded very seriously the guilt of innocent blood, sometimes viewed in terms of corporate responsibility. Like Pilate the priestly officials wanted nothing further to do with the situation, and likewise understand that the blood was innocent...
The money was profaned and tainted by the way it was used. By ancient thinking, it was ritually unclean (link 2 below) -- though even today a charity may refuse money if it is gained by ill-gotten means.
Now it follows that when they transacted the business of the field for the temple, to avoid association with ritual uncleanness, the priests would have to have bought it in the name of Judas Iscariot, the one whose blood money it was. The property and transaction records available to the public and probably consulted by Luke would reflect that Judas bought the field -- or else Luke is indeed aware of what transpired and is using just the right verb to make the point.
If Matthew meant to say that Judas died like the traitor, why didn't he just say so plainly?
He said so plainly, to his contemporaries. That modern fundamentalists think that nothing is said "plainly" unless it is said literally is an interesting insight into the fundamentalist mindset, but it does not change the realities of communication among Biblical peoples.
Matthew says the name 'Field of Blood' came because it was bought with blood money. Luke says it was because Judas split his guts all over. So which is it?
This objection assumes that what was "known unto all the dwellers" was Judas' gut-bust episode, but it would seem that the phrase modifies all that precedes it: "Now this man purchased a field with the reward of iniquity; and falling headlong, he burst asunder in the midst, and all his bowels gushed out."
Judas' gut-burst would hardly warrant a "field of blood" designation for the whole property. There would not be blood everywhere. The "Field of Blood" name was derived -- even as Matthew says -- from the act of purchase with the reward of Judas' iniquity -- what iniquity? The betrayal of innocent blood, which Luke recorded in his own Gospel.