The rapid advancement of biogerontology, particularly through interventions like senolytic therapies and stem cell rejuvenation, is transforming the prospect of radical life extension and even biological immortality from science fiction into a plausible future scenario. This impending paradigm shift necessitates a profound re-examination of existential psychology, which has traditionally posited awareness of mortality as the fundamental source of life's meaning, motivation, and authenticity. This article argues that the psychological phenomenon of death awareness would not simply vanish with the elimination of biological finitude. Instead, it would undergo a critical functional transformation. It proposes the concept of an "existential alarm clock," a reconstituted internal mechanism that shifts from serving as a chronological limit to acting as a qualitative regulator of existence. In a state of immortality, this alarm would awaken the individual from the unique perils of an endless lifespan: existential apathy, identity stagnation, and the "bad infinity" of undifferentiated time. The article analyzes the mechanisms of this alarm—triggering identity crises, combating profound boredom, and stimulating self-transcendence—and explores its vast social, cultural, and ethical implications. It concludes that, paradoxically, a continued dialogue with the concept of death remains a crucial condition for a meaningful and authentic life, even in the context of biological immortality.