It's like "the one" or "the soul mate" - no such thing, don't bother.
That may be the best answer to this question I've seen.
But I'll add my two cents, if the OP is interested:
Alpha male literally means #1 male. PUAs are interested in the alpha male because of the hypergamous female tendency to want to select the best possible mate to sire their child. Women are attracted to the "top dog", whether it be the best race car driver, the champion boxer, the strongest guy in the bar, the handsomest guy in the club, whatever. Women will seek out the favor of the top male, that's why they say the alpha male doesn't have to pursue women, he has his choice of a large number of options.
PUAs have tried to analyze the behaviors of "naturals" who have a lot of options when it comes to women, and they try to mimic these behaviors and attitudes. They call these attitudes the "alpha mindset".
The problem is, over time more and more people have added their own thoughts of what traits they think this "alpha male" should have, and what we have now is a mythological character who could not possibly exist in real life. They've piled so many different traits into this being that the concept is now essentially meaningless.
The "alpha male" is mainly only useful as a teaching tool, to teach attitudes and behaviors that women find attractive. Unfortunately, too many people take the concept too seriously, and try to place these perfect creatures in real life, which doesn't quite work. In our modern society, most alpha males are "situational". For example, a guy might be the top dog at work, but put him on a basketball court and he's an also ran. Modern life is too complex and varied for one male to dominate at everything he does, in every situation.