Amorgos and Mykonos are two of the most different islands in the Cyclades, and the direct boat between them is a summer-only Seajets service. Outside the April to October season there is no direct link at all, and even in summer there is often just one sailing on the days it runs, so this is a crossing to plan your dates around rather than one you can leave to the last minute.
Amorgos is long, steep and quiet, the island of the cliff-hung Hozoviotissa monastery, walking trails between old villages and deep blue coves. Mykonos is its busy, glamorous neighbour to the north-west, with the windmills and Little Venice, beach clubs and a nightlife that makes it one of the liveliest islands in Greece. Travellers often use this crossing to swap a week of hiking and swimming for a few nights of town life, or to finish a Small Cyclades trip near Mykonos airport.
The boats leave from Katapola, the main port of Amorgos. The island also has a northern port at Aegiali, and Ferryhopper lists Aegiali to Mykonos as a pairing, but it showed no sailings or fares for it when we checked. If you are staying in the north of Amorgos, plan on getting yourself to Katapola.
This is not a cheap hop. With a single high-speed operator, one adult cost €89.70 when we checked and two adults €179.39, and there was no car fare on offer. If you are on a budget, or travelling in the off-season, the practical answer is to go via Naxos or Paros, which both have boats from Amorgos and plenty onward to Mykonos.