Do the Mt. Everest rescue helicopters have modified engines to operate at high altitudes?
No, the helicopters are standard production versions. The Eurocopter AS350 is a common model used for these operations.

In 2005, Didier Delsalle landed a Eurocopter AS350 B3 on the summit of Mt. Everest at 29,029 feet (8848m) (twice). The only changes he made to the standard version were removing a few things like extra seats to reduce the weight, and of course flying solo. It also took careful planning and the use of updrafts to get there.

The B3 version of the AS350 (now called the H125) is marketed as high performance, designed for operation in hot and high conditions. It also holds the record for the highest long-line rescue mission at 25,590 feet (7800m) (in the Himalayas).

Although the official ceiling for hovering out of ground effect is 11150 feet (3400m), the flight ceiling is 23,000 feet (7000m), and rescues at 20,000 feet (6000m) are typically limited to 2 passengers rather than the full capacity of 6.

The Safran Arriel 2D engine used in the B3 version is rated up to 952 Shp on takeoff and operation up to 29,500 feet (9000m).