Samurai Champloo/WMG

Samurai Champloo is intended as a Lighter and Softer successor to Bebop
For one thing, the protagonists of Champloo seem to be able to access (often by stealing) decent food on a fairly regular basis, so their Perpetual Poverty isn't quite as bad as the Bebop's crew. Also, whereas Spike seemingly dies at the end of the series, Mugen takes Spike's survival skills further, and even when he does die in one case, he comes back.
 * Lighter and Softer? They cut off peoples limbs and kill many, many more people in much, much more graphic ways!
 * And we get to laugh about it. Most of the time.

Mugen and Jin still had some of that zombie-inducing wasabi, which they ingested shortly before the last episode.
This is the only way this troper can imagine how they survived.
 * Alternatively, the entire series after the part where they consume the mushrooms was an acid trip
 * Mushrooms don't give you acid trips.
 * Correct. Acid gives you mushroom trips.
 * They may not give you acid trips, but they'll make you see things, all right.
 * Turns out, Magic Mushrooms secrete LSD, or something so similar it may as well be.

Mugen was sexually abused/raped as a child.
He grew up in a poor, socially chaotic and violent environment while likely not having any protective and loving adult figure in his life. Places like that in the real world almost always have rampant child abuse of every kind, male or female. Around the end of the show, he makes a not-really-joking crack about how he'd been through every degrading, humiliating thing possible...

Shoryu is an Airbender

 * He learnt how to use his chi from monks on the mainland, and he does so by shooting wind blasts from his sword.
 * Except Airbending is chi-powered, not chi itself.
 * In Aang's series, various benders learn how to manipulate new bending techniques with different elements from their designated powersets. For example, Katara learns to bend blood, Toph with Metal and Azula with lightning. These new technicques are STILL derivatives of pure chi, that's a given, but the ability to use pure chi was never out of the question either.

Mugen is a distant ancestor of Spike.
Same voice. Same hair. Do the math.
 * Even better: both series are done by the same guy (Sinichiro Wanatabe) and in fact Samurai Champloo is a spiritual sequel to Cowboy Bebop (as much as Cowboy Bebop was an homage to bebop jazz, hence the title, Samurai Champloo is an omage to modern rap-inspired bebop, hence the title, "Champloo" - Japanese for "remix") - or perhaps an outright prequel?
 * Better still: Spike and Mugen were voiced by the same actor, Steve Blum, in their respective English dubs. Coincidence? I think not!
 * If it's going by the afro-like hair, they'll have an unpleasant family reunion with Nabeshin, where he'll disappear to another mission, not before drinking all their booze and eating all the goods, because he's not the kind of guy to lose a good chance.
 * Sorry, but Watanabe Shinichi (Nabeshin) is a different person. He and Watanabe Shinchiro who directed this show just have very similar names, but Shinchiro doesn't have an afro or Lupin III inspired outfits.
 * Plus, how many times have they both been mortally wounded but pulled through?

The entire series is a game played by the crew of the Bebop.
Mugen, Gin and Fuu really ARE Spike, Jet and Faye, only playing some sort of MMO or PnP game.
 * And they've all made characters that reveal much of what they think of themselves. Spike plays a badass martial artist, Jet plays a hardass law & order type and Faye plays an innocent girl.
 * Further, the reason for the series inexplicably bringing in zombies or baseball was just Ed playing with the system.
 * She's the GM, Ein is playing as Momo.

Mugen is immortal.

 * He's been killed three times, and he's still alive.
 * Specifically an immortal of the Highlander series.
 * "Mugen" means "infinity", so yeah I'm going with this one.

Mugen IS Spike.
Adding onto the immortality entry above: "There once was a tiger striped cat. This cat died a million deaths, revived and lived a million lives, and he was owned by various people who he didn't really care for. The cat wasn't afraid to die. Then one day the cat became a stray cat, which meant he was free. He met a white female cat, and the two of them spent their days together happily. Well, years passed, and the white cat grew weak and died of old age. The tiger striped cat cried a million times, and then he died too. Except this time, he didn't come back to life."

Mugen, Jin, and Fuu are all the ANCESTORS of each of the three main leads of Cowboy Bebop

 * Why not? It's been nearly 200 years. Plenty of time for the three of them to have married and produced children, who also married and produced children...and for some of those descendants to have met and married each other unknowingly. Spike and Fay are both a little crazy like Mugen, Spike has Mugen and Jin's combat skills (albeit with firearms and a spaceship), Jet has some of Jin's calm (but can go off like Mugen if angry enough), and all three have Fuu's uncanny ability to meet -- interesting -- people.

A time traveler was involved

 * A hip hop loving time traveler went back in time and taught the Japanese about hip hop.