There is no complimentary hotel shuttle service to and from the airport. 

Canyon Transportation is the resort’s  preferred vendor for shuttle service to/from the hotel, and they provide discounted pricing.  Reservations can be made online up to 24 hours before the desired service here or by phone (801) 255-1841. When making your reservation, make sure to mention that you are a guest of Zermatt. Please see the attached rate sheet from Canyon transportation for pricing information. 

Alternatively, a Taxi ride is in the range of $100 -$120) and Ubers/Lyfts are plentiful at the airport, and would run about $60 from the Airport to Zermatt. Travel time is just under one hour.  Upon your departure, keep in mind that getting an Uber/Lyft driver usually takes about 10-20 minutes before arrival.