All right, then. From your location and the description of what the guy did, I'm reasonably sure that it was some form of Chinese martial arts. There's similar stuff in Southeast Asian and Indian forms of martial arts, but they aren't nearly as popular in the Americas.

CMA is not my best area, but here's a listing of some Chinese martial arts ordered into three categories: those that most closely match what you've described, those I can't rule out, and those I'm pretty sure don't match. Try contacting schools that teach the relevant styles in your city.

Best bets:

Choy Li Fut/Cai Li Fo (limb-and-head attacks, open-handed strikes with a wide turning motion of the whole upper body are common. See here and here for examples.)

Southern Praying Mantis/Chow Gar/Chu Gar/Kwong Sai Jook Lum (see here

Piguaquan/piguazhang (does simultaneous two-handed limb-and-head attacks, wide open-handed strikes with a turning motion of the whole upper body are common.)

Fujian White Crane (limb-and-head attacks, open-handed strikes with a wide turning motion of the whole upper body are common. See here and here for examples.)

Cannot rule out:

Wing Chun/Ving Tsun/Wing Tsun (commonly practiced, has limb-and-head attacks, but its movements tend to be more compact than what you've said in the thread)

Choy Gar/Caijia Quan (see Wing Chun)

Fut Gar (see Wing Chun)

Ng Gar/Five Family (see Wing Chun)

Hung Fut (see Wing Chun)

Hung Gar/Hung Ga/Hung Kuen (has some hand strikes of the type you've mentioned. Orthodox Hung Gar would strongly emphasize low stances, which you haven't mentioned, but wide stances tend to be the first thing to go when adapting a traditional martial art to practical use)

Jow Gar/Jow Ga (see Hung Gar)

Lama Pai/Hop Gar/Tibetan White Crane (has two-armed simultaneous limb-and-head attacks and wide, swinging hand techniques, but usually done with a closed fist)

Bak Mei/Pak Mei (two-armed simultaneous limb-and-head attacks are known, but the movements are generally more compact than you're describing and straight punches/elbows/backfists are more common than open-handed strikes)

Li Gar/Lee Gar (see Bak Mei)

Mok Gar (see Bak Mei)

Five Ancestors (this bit looks something like what you've described, two-handed simultaneous head-and-limb attacks are known- but FA is rarely taught martially, and movements are more compact than what you describe)

Low probability:

Northern Shaolin-type styles: Luohanquan, Meihuaquan, Changquan, Huaquan, Mizongyi, Tien Shan Pai, Tongbeiquan (not many simultaneous two-handed limb/head attacks, rarely trained martially)

Bājíquán (strikes are generally single, short-ranged straight punches or elbows to the body)

Northern Praying Mantis (you haven't mentioned its distinctive hand position, or swiftly-executed locks/grab-and-attack attempts, or extreme rapidity of techniques)

Eagle Claw (also principally known for forms of attack you haven't mentioned)

Shuai jiao (wrestling style)

Baguazhang (you haven't mentioned circular footwork, not usually practiced martially)

Taijiquan (very uncommon to practice martially, usually a grappling style when it is practiced martially)

Xingyiquan (swinging left-to-right strikes practically unheard-of)

Jeet Kune Do (looks more like kickboxing than traditional stuff, generally prefers straight punches from the outside)

Not mentioned:

Many Chinese martial arts that are rarely practiced, or are rarely practiced martially, in the Americas