Calling the Ju-88 a heavy day fighter is like calling the Beaufighter a fighter. It was really an attack platform. You wouldn't call the IL2 a fighter, right?
IMO, Probably the 109. Not just for sheer modifications to the airframe like some are listing here, but because it was used everywhere for everything.
It started as a simple continental fighter. Moved on to be used over Britain, the Med, in Africa, the middle east, the Soviet Union. It had everything from all MGs to MGs removed and only cannons, to no cannons and cameras. They even had carrier-capable variants (never mind that the carriers never developed). the airframe was not the best, but it was used everywhere and for everything. One version even carried a bomb so big they had to stick and extension tail gear under the fuselage to lift it up. Once airborn thise parachuted down. The plane dropped the bomb and landed with its normal tail gear. The plane was responsible for most of the RAF losses on the western front for a long time, even after Fw190s started showing up.
Of course it had its major failings. At first range was too short without a drop tank (rather, it was too short for the military doctrine of the BOB), and armament was always an issue after the E, hence the additions of higher-caliber nose cannon and wing gondolas, but these wouldn't be the same as other aircraft.
I think that many planes fit the bill, definitely the P-47 ranks up there as the plane that broke the back of the Luftwaffe and wiped the ground with the IJA in the pacific. I think the P-51 counts if you include its post-war record but during WW2 it was mostly just used for long range escort (unless you count the limited allison ground attack versions?). Then there was the bomber used extensively for fighter sweeps, the P-38 (*ducks, runs, hides*).
Overall, lots of choices, but out of stupid blindness, out of limited options, out of political restrictions, and out of the utter non-flexability of the german war machine industry, they just used that 109 for EVERY thing they needed. Not necessarily the best at every role, but I'd think the most versatile for sure.