A very good option these days is getting a Micro 4/3 Camera in an underwater housing.
Right now the cameras on that category that are leading the market and very popular for underwater use are the Panasonic Lumix DMC-GX7, Olympus OM-D E-M1 and Sony alpha NEX-7.
It's recommended to get them with a good wide-angle lens such as the Panasonic 8mm fisheye or the Panasonic 7-14mm, and you can also get the Olympus 60mm for Macro. For Sony you can get the Sony E 16mm f/2.8 "Pancake" lens.
These cameras have fantastic performance above water, and equally superb performance underwater. Using a housing by Nauticam, Aquatica or other manufacturers, attaching a strobe and perhaps a video light, you can achieve professional results and produce outstanding photos and videos.
Should you prefer to go with a dSLR system, remember that quality is very good, but prices are higher, the gear is heavier and harder to travel with, and you need more experience in diving and photography to get good results. That's why a mirrorless system is best to start from.
Going with a compact camera is also a good option, as today compact such as the RX100 Mark II and the G1X Mark II achieve great quality and functionality in a very good price and compact size.
Pairing them with a good strobe would also produce great results.
Here are some links to the mirrorless camera's reviews:
http://www.dpreview.com/reviews/olympus-om-d-e-m1
http://www.dpreview.com/reviews/sonynex7
http://www.dpreview.com/reviews/panasonic-lumix-dmc-gx7
And the relevant housings:
http://www.housingcamera.com/Nauticam-Housing-4-Panasonic-GX7-p/na-17708.htm
http://www.housingcamera.com/Nauticam-17403-NA-NEX7-housing-for-Sony-NEX-7-p/17403.htm
http://www.housingcamera.com/Aquatica-Housing-4-Olympus-OM-D-E-M1-p/aq-32000-opt.htm