Alaska is something of an oddity in world politics- it's notable for being the only foreign colony ever controlled by Liechtenstein, brought by Prince Johann II from the Russian Tsar in 1867. At first the world press considered the move laughable, reporting it as 'Johann's folly', but the Prince was determined to make the territory profitable. He had the city of Johannesburg set up on the southeastern corner in the late 1870s and began trading with neighboring British Columbia, benefiting financially from the influx of white Anglo-Saxons who migrated to it, particularly once the gold rush began in the 1880s. On top of that, Alaska gave a foothold to the Austro-Hungarian and German empires in North America that previously had not existed, with many new migrants being German-speaking (though English as a second language was a priority due to its close proximity to English Canada and the US).
As one might imagine, though, indigenous people who had under Russian rule been relatively undisturbed (at least compared to those in the US and Canada) did not do well out of this arrangement and lost large amounts of their money and influence, and activism to protect their lands has been a major issue in Alaskan politics for more than a century. By the time World War I came to a close, however, the most pressing issue for white Alaskaners was that they felt greatly isolated from Vaduz because of the massive geographic distance, the lack of resources thanks to American and Canadian boycotts, and because there were by this point over 7 times as many Alaskaners as Liechtensteiners (around 64,000 compared to about 8,800) who were only represented by 3 indirectly elected members of the Liechtenstein Landtag, less than either the Oberland or the Unterland of continental Liechtenstein. Consequently, with Alaskans threatening a unilateral declaration of independence or even a revolution if home rule was not granted, as part of the 1921 Constitution, Johann II allowed the formation of the Landtag of the Principality of Alaska, giving Alaska home rule within Liechtenstein.
Alaska's first Prime Minister, despite the predominantly German-speaking white population, was a Canadian journalist from New Brunswick who had moved to the territory in the late 1890s- John Franklin Alexander Strong. Strong was a colorful politician to say the least- a bigamist and ex-newspaper magnate who had advocated for home rule, he offered citizenship to indigenous people who gave up tribal life, but retained popularity for both himself and the Patriotic Union party he led by implementing workers' compensation and old age pensions. One thing that ended with Strong, however, was that every leader of Alaska since has been a German-speaker.
During the run-up to World War II, Alaska also became a hotbed for German refugees under threat from the emerging fascist governments of Germany and Austria, particularly leftists like social democrats (though Alaska's government did little to facilitate communist or Jewish refugees). One of the most prominent of these was former Minister President of Prussia Otto Braun, who formed the SPA (Sozialdemokratische Partei auf Alaska) as a rival to the existing centre-right Progressive Citizens' Party and centrist PU to create a welfare state in the model of FDR's New Deal. It worked out very effectively, with Braun building up sizeable state infrastructure during his three terms, but his party lost re-election in 1947 and the emergence of the Cold War split the SPA between anti-communists like Braun and pro-communists unafraid of the US government trying to seize control of the territory, allowing the PCP and PU to unify against the left and keep them out for decades to come.
While the leadership of Alaska was held by PCP figures from 1947 to 1994, this papered over a much more fractious situation with the opposition throughout this time, with a diverse series of politicians from Ernest Gruening, who sought in vain to reunify the SPA, to the libertarian anti-war figure Maurice Gravel, who fought to keep Prime Minister Elmer Rasmuson sending Alaskaner troops into Vietnam, to Johannes Coghill in the 1980s, whose Alaskan Independence Party (AKIP) advocated for the party severing ties with Liechtenstein.
It was AKIP's ascendancy that finally started to crack the PCP's dominance, as they ditched the PU and allied with Coghill basically to give Walter Hickel a conservative majority against the increasingly uncooperative and liberal-conservative PU. This backfired spectacularly, as the AKIP fell out with the PCP due to their refusal to budge on Alaskaner independence, and Anthony Knowles, an emigrant to Alaska and mayor of Ankerplatz* during the 1980s, managed to build up grassroots support for the PU and force the PCP out of government for the first time in 47 years. The PU was accompanied by a new left-wing party- the Inuit-Freie Partei (Inuit Free Party), a catch-all leftist group that advocates for Inuit rights, green politics, social democracy and pretty much anything else the PCP doesn't countenance.
Once Knowles left office in 2002, though, the PU's new leader Frank Murkowski moved the party back to the centre, and in 2006 helped make Sarah Palin of the PCP the first woman elected Prime Minister of Alaska after the PCP surpassed the PU in votes. This did not go well for either party, as Palin garned the nickname 'der Alaskaner Merkel' for seeming dictatorial and unlikable and the left deserted the PU in droves. After Palin lost a motion of no confidence in 2009 at the height of the Great Recession, Alaska gave a plurality of seats to the IFP under Markus Begich for the first time ever, but Begich also ran into problems, losing popularity among Alaskaners on the right for his interventionist economic policy and on the left for being considered too moderate.
The precedent of one-term Alaskaner PMs continued in the 2013-18 term, one of the longest in recent history, as with the IFP routed and the PU and PCP winning 15 seats each, independent Wilhelm Walker came through the middle and effectively stayed PM of a grand coalition for five years because he managed to be inoffensive enough to the parties in the Landtag to win re-election, though it was clear he had no chance of getting a second term and to try and save himself even admitted he would not be a candidate for the office again. He lost his seat in Pedroni Ost** to the PU along with four of the six independents who had been elected in 2013 (and, probably not coincidentally, the ones who had been part of his cabinet), while the IFP and PU each gained a seat and the PCP gained two.
As a result of this, PCP leader Cathrin Giessel declared that she would seek to become Prime Minister since her party had a plurality of seats, but wary of the decision of her father, PU leader Lisa Murkowski refused to support Giessel as leader. Instead, she agreed a confidence and supply arrangement with the leader of the IFP Byron Edgmon, making him the first indigenous person to serve as Prime Minister of Alaska.
This calculated move seems to have worked a treat for Murkowski, as the PU have surged in the polls and avoided much of the fallout for any unpopular policies Edgmon has implemented. While Murkowski has asserted she has no plans to try and unseat Edgmon, he has made himself unpopular with the IFP for allowing the PU to shoot down a number of its policies during the 2018-23 Landtag session, though his approval ratings have risen during the COVID-19 pandemic as he has been able to make the PU acquiesce with him for the greater good.
While Giessel has managed to hold onto power within the PCP, the party's strength has been falling since she failed to become PM, and it seems probable the main battle come 2023 will be which one of Murkowski and Edgmon gets to rule the roost.
(I just realized partway through this this would make sense of why Alaska isn't part of the American Federation in that TL, so you can probably consider this a part of that if you like, or just a random what-if-Liechtenstein-bought-Alaska-when-offered-it scenario if you prefer.)
*OTL Anchorage, literal translation
** OTL Fairbanks, named for early settler Felix Pedroni