Law and Chaos might have no relevance in some campaigns. DMs may decide to forego alignments completely, or create an alternate system. A new system should include a Neutral or Unaligned option, and at least one active alignment.
Monopolar alignments contrast with Unaligned, but have no true opposite. Spells detect merely its presence or absence. Examples in gaming include Shadow from Midnight and Chaos from Warhammer. One could also make this force positive, like Gnosis, a knowledge of how the world really works. An alignment toward the "Uncanny" would imply an affinity with magic, faerie, the supernatural, or however one describes it; it's neither good or evil, but it might upset psychically sensitive folks and animals.
Bipolar alignments work similarly to Law/Chaos: Good/Evil, Man/Nature, Sky/Earth, Matter/Spirit, just about any duality that a GM can clearly articulate and players can understand.
The "Good"/"Evil" axis presents one major difficulty: when does a being register on a Detect Evil spell? Thinking bad thoughts? Commiting a crime? How bad a crime? Ramifications of such definitions have fueled some very long threads, so it's simpler to treat Good and Evil the way LotFP treats Law and Chaos: Good characters consecrated themselves to the Forces of Good, and Evil characters have sold their soul to the Forces of Evil. Only clerics, paladins, and angels detect as Good, and only anti-clerics, anti-paladins, and demons detect as Evil. Good seeks universal peace and happiness, Evil seeks an eternity of suffering and fear; petty theft and acts of charity change nothing. Clerics and paladins (and their opposites) can never deny or escape their role in the Great War, so a drunken whoring ex-paladin might suddenly get his powers back just as demons descend on his town.Multipolar alignments might follow any theme: the elements (three Hindu, four Western, five Taoist or Buddhist), the colors of Magic: the Gathering, the principles of Creation, Preservation, and Destruction, or just about anything else meaningful. Each should be independent and atomic, although some might oppose each other more strongly: Fire and Water are mortal enemies, but Water and Earth might form an alliance of convenience.
Note, however, there's no point in alignments unless they have some meaning in the game world. By this I mean more than spells and game mechanics. If a campaign doesn't revolve around Good and Evil, if both aren't palpable forces in the cosmos, then don't use them. Think of Chekhov's gun: if you introduce Good and Evil in the beginning, you'd better use them before the end.
The Temporal Alignments
Here's an alignment system I've been toying with. In addition to the standard OD&D three, there are three others representing three factions fighting over the flow of time.
Lawful, Neutral, Chaotic: See the LotFP definitions.
Entropic: Entropy devours all. Time grinds mountains into dust, dust into atoms, atoms into brief burst of light. Only a fool opposes the inevitable decay of all things. Chaos understands change, but deludes themselves into thinking their dance will continue forever. History tells itself stories to make sense of time, but it knows where the story ends. Law and Flux spread false hopes of escaping oblivion; only by accepting one's doom can one find peace.
Fluid: Time is always in flux. Only the eternal Now exists; the past is a story, the future unwritten. Blink and the world can change completely, quite literally, and the old world is a half-remembered dream. Sometimes the new world makes sense, sometimes it's cruel and random, but there's always the next one. Law and Chaos can play their games; to be free one must shed fear of the future and chains of the past.
Historic: Time is a chain of causes and effects, not always predictable at the time but sensible in hindsight. Entropy is but the tick of the clock, Law but the perception of the future implicit in the past. Chaos is but order unrecognized, and Flux the delusions of poets and mystics.
Imagine them arranged on a five-pointed star. Neutral lies in the center. On each point, clockwise, sit Fluid, Chaotic, Entropic, Historic, and Lawful. Points next to each other sometimes ally against a common enemy, but each point is an implacable enemy of the two opposite points.
So great, but what's the point? The notion of a Time War and time travel as a weapon crops up in a lot of science fantasy: Doctor Who, Fritz Leiber's The Big Time and related stories, Bill and Ted's Excellent Adventure, and the infamous Temporal Cold War plotline in Star Trek: Enterprise. In gaming one finds Super Genius Games' Time Thief and Time Warden classes for Pathfinder and all manner of time-travel RPGs. Add to the mix John Wick's "The Flux", a "meta-RPG" in which players shift universes, characters, and game systems while pursuing the same goals and facing incarnations of the same antagonists.
So, imagine among the pseudo-medieval and gothic shenanagans three powers vie for the future and past of the world. The Historic faction wants a clean timeline with no paradoxes or loose ends, employing academics and time-hopping Men in Black alike. The Entropic faction are nihilists who know the cosmos will end with a whimper and a gulp, and just want to cut to the chase. The Fluidic faction sees Entropy chomping and History dragging out the inevitable; its outside-the-box strategy involves shifting between universes to find a solution, or failing that escape.
The PCs stumble upon this struggle while killing things and taking their loot, probably by experiencing the Flux for themselves. In the process they face Time Wardens of History and Corruptors of Entropy (whatever they end up being). Maybe the PCs gain Time Thief allies, or new/replacement PCs choose that class.
Maybe this is too high concept for an Old School D&D game, but some AD&D players killed gods and took their stuff back in the day. This Time War campaign doesn't rely on characters reaching high levels, since opponents are either humans with gimmicks or godlike beings with limitations. Players need only venture through gates to other times and worlds and follow clues to stopping the apocalypse ... which is pretty Old School in my book.