"medieval chemistry; the supposed science of transmutation of base metals into silver or gold" (involving also the quest for the universal solvent, quintessence, etc.), mid-14c., from Old French alchimie (14c.), alquemie (13c.), from Medieval Latin alkimia, from Arabic al-kimiya, from Greek khemeioa (found c.300 C.E. in a decree of Diocletian against "the old writings of the Egyptians"), all meaning "alchemy," and of uncertain origin.
Perhaps from an old name for Egypt (Khemia, literally "land of black earth," found in Plutarch), or from Greek khymatos "that which is poured out," from khein "to pour," from PIE root *gheu- "to pour" [Watkins, but Klein, citing W. Muss-Arnolt, calls this folk etymology]. The word seems to have elements of both origins.
Mahn ... concludes, after an elaborate investigation, that Gr. khymeia was probably the original, being first applied to pharmaceutical chemistry, which was chiefly concerned with juices or infusions of plants; that the pursuits of the Alexandrian alchemists were a subsequent development of chemical study, and that the notoriety of these may have caused the name of the art to be popularly associated with the ancient name of Egypt. [OED]
The al- is the Arabic definite article, "the." The art and the name were adopted by the Arabs from Alexandrians and entered Europe via Arabic Spain. Alchemy was the "chemistry" of the Middle Ages and early modern times, involving both occult and natural philosophy and practical chemistry and metallurgy. After c. 1600 the strictly scientific sense went with chemistry, and alchemy was left with the sense "pursuit of the transmutation of baser metals into gold, search for the universal solvent and the panacea."
Middle English neue, from Old English neowe, niowe, earlier niwe "made or established for the first time, fresh, recently made or grown; novel, unheard-of, different from the old; untried, inexperienced, unused," from Proto-Germanic *neuja- (source also of Old Saxon niuwi, Old Frisian nie, Middle Dutch nieuwe, Dutch nieuw, Old High German niuwl, German neu, Danish and Swedish ny, Gothic niujis "new").
This is from PIE *newo- "new" (source also of Sanskrit navah, Persian nau, Hittite newash, Greek neos, Lithuanian naujas, Old Church Slavonic novu, Russian novyi, Latin novus, Old Irish nue, Welsh newydd "new").
From mid-14c. as "novel, modern" (Gower, 1393, has go the new foot "dance the latest style"). In the names of cities and countries named for some other place, c. 1500. Meaning "not habituated, unfamiliar, unaccustomed," 1590s. Of the moon from late Old English. The adverb, "newly, for the first time," is Old English niwe, from the adjective. As a noun, "that which is new," also in Old English. There was a verb form in Old English (niwian, neowian) and Middle English (neuen) "make, invent, create; bring forth, produce, bear fruit; begin or resume (an activity); resupply; substitute," but it seems to have fallen from use.
New Testament is from late 14c. New math in reference to a system of teaching mathematics based on investigation and discovery is from 1958. New World (adj.) to designate phenomena of the Western Hemisphere first attested 1823, in Lord Byron; the noun phrase is recorded from 1550s. New Deal in the FDR sense is attested by 1932. New school in reference to the more advanced or liberal faction of something is from 1806. New Left (1960) was a coinage of U.S. political sociologist C. Wright Mills (1916-1962). New light in reference to religions is from 1640s. New frontier, in U.S. politics, "reform and social betterment," is from 1934 (Henry Wallace) but associated with John F. Kennedy's use of it in 1960.