Why did France put so many resources into fighting Algeria in comparison to its other former colonial holdings? Was there anything unique about the French-Algerian relationship. On Wikipedia it says that France regarded Algeria in a way similar t how the USA think of Hawaii and Alaska. Is this true? If it is, why?
Algeria was more then a colony, there were French Départements in Algeria, from 1848 on until 1962 it was an integral part of the French motherland.
This does not mean, that Algerians were full citizens.
See also Process of Colonization:
Algeria was formally declared to be a part of France and was again divided into three departments, the government of which was placed into the hands of European settlers.
The tension between the French and the Arabs started to grow. The French neglected the Arabs and in July of 1864, Senatus Consulte declared all Moslems to be French subjects, meaning the Muslims were not to be citizens. This way, the Arabs could keep their Muslim religion and the Koran law, and were subject to special police regulations, including the infamous code de l’indigenat. In order for Arabs to become full-blooded French citizens, they had to give up their personal status, which not many were willing to do.
The other answers pointed out very correctly that Algiers was legally considered a part of France. What must be added is that there was a very large body of French colonists in Algiers, which was not the case with the other French possessions, whatever their legal status.
Julian Jackson's account of France in the 1930s and 1940s France: The Dark Years shows how France's African empire was seen as compensation for the decline in French prestige and influence in Europe during this period.
Algeria was the most important element in this project because of proximity to France, long historical association, and larger settler population than elsewhere. Writing in 1938 Le Matin declared the keys to French security were the Vosges, Alps, Pyrenees and Atlas Mountains (my emphasis).
Le Matin also stated at this time:
France has two capitals: Paris and Algiers
It's hard to imagine an equivalent statement of imperial projection coming out of London or other European capitals.
Algeria was considered a "back door" to France, from the south. In the hands of a hostile country, England, Germany, or even Libya/Egypt, it was seen as a potential invasion route to France. On the other hand, if France retained a foothold in Africa through Algeria, it could continue to influence its former colonies in "French West Africa" (which it does to a small degree, to this day).
France looked at Algeria that way an American might see Latin America through the Monroe Doctrine: a "no fly" zone for hostile foreign countries. Algeria is the part of Africa that is closest to France.
There were multiple reasons why Algeria was so important to French:
Algeria was French military colony since 1834. and, by the constitution of 1848 to be an integral part of French territory and divided into three French departments (Algiers, Oran and Constantine). Also, as Tom said, Algeria was backdoor of France and it was very possible route to invade France.
French capitulation in WW2 was already great hit to French honor, but when France lost in First Indochina war France was totally defiled. France could not allow to lose in another war because that could cause big political and military crisis in France.
In Algeria, there were more French population (about 800,000) then in other overseas territories and colonies and their destiny would be very critical if they lost that war.