Altaïr's Assassin stronghold in Assassin's Creed is Masyaf.
Masyaf is a fortified settlement in present-day Syria and serves as the headquarters of the Assassin Brotherhood in the first game. Altaïr returns there between missions to receive guidance, report to his mentor, and learn how his actions affect the Brotherhood.
The game presents Masyaf as a dramatic mountain fortress surrounded by a settlement. Its fictionalized role draws on the historical association between the medieval Syrian fortress of Masyaf and the Nizari Ismaili community, often called the Assassins in popular Western histories.
Alamut is a different fortress, located in modern-day Iran and also strongly associated with the historical Nizari Ismailis. Acre is one of the game's major mission cities, while Constantinople becomes important in Assassin's Creed: Revelations, so neither is Altaïr's home base.