  A motivation of using noncommutative and nonarchimedean geometry on very
short distances is given. Besides some mathematical preliminaries, we give a
short introduction in adelic quantum mechanics. We also recall to basic ideas
and tools embedded in q-deformed and noncommutative quantum mechanics. A rather
fundamental approach, called deformation quantization, is noted. A few
relations between noncommutativity and nonarchimedean spaces as well as
similarities between corresponding quantum theories on them are pointed out. An
extended Moyal product in a proposed form of adelic noncommutative quantum
mechanics is considered. We suggest some question for future investigations.
