Hunting is a symbolic search for something needed, but "unknown".
Since the dawn of human history animals have been the most potent symbols for the hidden aspects of a person. So it often works out that (psychologically) the hunter is in search of an animal that offers some quality that the hunter lacks.
Among primitive people the power of an animal is consciously assimilated by killing the animal and eating it, thus hunting is often associated with killing.
The difference between hunting and stalking is important. If you know what you are pursuing, then you are stalking, not hunting. Hunting provides the psyche with some new, formerly unknown element. Stalking is an attempt to recreate some past successful hunt. Stalking isn't necessarily bad, but it is not hunting.
Most cultures have local examples of an archetypal hunter, one who hunts in a way that is exemplary and serves to teach the process to others (through legends etc). But the real archetypal hunter, the one that would be common to all humans, is himself (or herself) a mysterious animal not yet caught. The reason being that the archetype has to be internal and real rather than academic in order to actually exist. In other words the hunter that exists outside of you is of little use to you.
Here is an interesting way of seeing hunting.
There is a popular online test for Asperger's Syndrome, a form of autism. The results for the test list two approaches to hunting, one for "aspies" (those with Aspergers) and one for "normal" people. Two ways of looking for something unknown.
Asperger's Hunting. This group contains passive hunting traits. One part of the traits is related to preferred habitats (e.g. slowly flowing water; caves; woods; liking mist or fog). Another part seems to be close-contact hunting traits (e.g. jumping over things; climbing; chasing animals; biting; enjoying spinning in circles; strong grip; strong hands; physical endurance; enjoying rodeo riders). Some other traits are related to sneaking (e.g. sneaking through the woods; sneaking up on animals; walking on toes) and general hunting tactics (e.g. mimicking animal sounds; digging; throwing small things; building traps; fascination for fire; sniffing)
Normal hunting. The traits in this group are related to cooperative hunting. These traits are often described in terms of dysfunctions. Typical traits are recollections of environmental information (e.g. positions of things; scores in games; order of words, letters and digits; map reading) and passing on information to others (e.g. passing on messages; knowing left from right; dates and times of events; remembering appointments and events; reading clocks and calendars; carrying over information between contexts). Other traits are related to trading and exchange with others (e.g. calculating change from a purchase; knowing what to bring to appointments; remembering sequences of past events; remembering formulas; filling out forms).
The hunter archetype is related to the concept of Wu Wei in Taoism. "Hunter" describes a natural human aspect, not a contrived activity. The point being that before there was any interest in studying the hunt, the hunter was hunting. After the (intellectual) interest developed, the hunt was not improved.
The Wikipedia page on Orion has some academic history on the development of perhaps the most famous archetypal hunter.
"When a particular flower blooms, that tells us it is time to collect a certain type of plant or hunt a certain animal," says Wanyubi (an Australian Aborigine).
What was lost when people came out of the forests and animals went from being neighbors to being objects?