By alpha damage I mean the max damage that a weapon can initially "do" without any gems placed into it, this means bonuses are okay. I ask this because I saw someone who has a weapon that does about 8,000 damage without any gems placed in it.
Check out http://www.d3maxstats.com/
It displays the maximum damage of each weapon type at level 70.
It is worth noting that no weapon in the game will ever have 8000 tooltip damage.
As has been mentioned, weapon damage is a small part of sheet DPS, which is in turn a small part of actual DPS. In practice, you simply use the best weapon for your build which drops. However, for those who like theorycrafting, here is what I have been able to dig up from http://us.battle.net/d3/en/item/. It is consistent with the d3maxstats link above, but is from a primary source.
The way the question is worded is a bit odd, so I'm going to answer two questions:
What is the max damage that a weapon can display?
The maximum damage available on any normal level 70 legendary is 3700. This is found on two-handed maces. Ancient Legendaries should be able to roll 30% more damage, bringing this number up to ~4810. There is little concrete data available about ancient legendaries at this time, so it is possible this number is slightly off. The highest I was able to find with a google image search is 4529.
What is the maximum tooltip damage that can be displayed?
Tooltip damage is equal to the average damage on the weapon multiplied by the attackspeed of the weapon (while not wearing any other gear, or using passives/active abilities) multiplied by the %damage boost of the weapon. Assuming you roll max damage(1788/2324.4 on Ancient Legendaries), max attackspeed(7%), and max %damage(10%), the max damage of any weapon goes to Monk Diabos at 4105.317 DPS for a normal legendary or ~ 5336.912 DPS for an Ancient legendary.
The problem with what you are asking is you are singling out a single piece of gear on a pure stat based game completely Dependant on RNG to figure your stats.
Diablo 3 uses RNG is figure out how awesome your states are. Currently there are no Weapons out there that do 8k Weapons damage(this is the Aplha stat you seek, but we all know it as just Weapons Damage).
Currently there is a thread on Battle.net that has people reporting no less than 5k Weapons Damage on Ancient Prefixed Items. What I have been to verify is This weapon as currently highest I have seen.