The Collaborative International Dictionary of English v.0.48:

Hypostasis \Hy*pos"ta*sis\, n.; pl. Hypostases. [L., fr. Gr. ?
   subsistence, substance, fr. ? to stand under; ? under + ? to
   stand, middle voice of ? to cause to stand. See Hypo-, and
   Stand.]
   1. That which forms the basis of anything; underlying
      principle; a concept or mental entity conceived or treated
      as an existing being or thing.
      [1913 Webster]

   2. (Theol.) Substance; subsistence; essence; person;
      personality; -- used by the early theologians to denote
      any one of the three subdivisions of the Godhead, the
      Father, Son, and Holy Spirit.
      [1913 Webster]

   Note: The Council of Alexandria (a. d. 362) defined
         hypostasis as synonymous with person. --Schaff-Herzog.
         [1913 Webster]

   3. Principle; an element; -- used by the alchemists in
      speaking of salt, sulphur, and mercury, which they
      considered as the three principles of all material bodies.
      [1913 Webster]

   4. (Med.) That which is deposited at the bottom of a fluid;
      sediment.
      [1913 Webster]

WordNet (r) 3.0 (2006):

hypostasis
    n 1: the suppression of a gene by the effect of an unrelated
         gene [syn: hypostasis, epistasis]
    2: the accumulation of blood in an organ
    3: any of the three persons of the Godhead constituting the
       Trinity especially the person of Christ in which divine and
       human natures are united [syn: hypostasis, hypostasis of
       Christ]
    4: (metaphysics) essential nature or underlying reality

Moby Thesaurus II by Grady Ward, 1.0:

72 Moby Thesaurus words for "hypostasis":
   abstract noun, adherent noun, air, atom, atomic particles, axiom,
   brute matter, building block, center, chemical element,
   collective noun, common noun, component, constituent, core, earth,
   element, elementary particle, elementary unit, elixir, essence,
   essential, fire, flower, focus, fundamental, fundamental particle,
   gerund, gist, gravamen, heart, hyle, inner essence, kernel, marrow,
   material, material world, materiality, matter, meat, molecule,
   monad, natural world, nature, nominal, noun, noun phrase, nub,
   nucleus, nuts and bolts, physical world, pith, plenum, postulate,
   principle, pronoun, proper noun, quid, quiddity, quintessence,
   quotation noun, sap, soul, spirit, stuff, substance, substantive,
   substratum, the four elements, the nitty-gritty, unit of being,
   water