The best replacement for adieu is ado.

The best alternatives to adieu are rated as following:
★★★★★  ado
★★★★  au revoir
★★★  adios
★★  adieu
★  without further ado
★  adieux
  should it be ado

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Speaking about adieu, one probably shouldn't forget about further.

Other search terms related to adieu and their ratings are:
★★★★★ further
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Google Trends

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adieu
ado
au revoir