Mercedes driver Hamilton clinched his fifth drivers' championship during last month's Mexican Grand Prix.
Dutchman Verstappen has won two races this year and was cruising towards another in Brazil on Sunday before a crash with Esteban Ocon ruined his hopes.
Hamilton, who took victory in Sao Paulo, has been impressed with Verstappen this year and expects the 21-year-old and his Red Bull team to provide serious competition in 2019.
"This year they've been incredibly strong and I can only imagine next year, they're going to be even stronger," Hamilton told Sky Sports.
"I think for the sport it will be great if Red Bull are more in the battle with Ferrari and Mercedes and we welcome any more teams.
"Max is driving really well this year.
"He's such an awesome, young talent. I'm looking forward to many races with him.
"At the weekend in Brazil, we had the engine problems so I didn't feel I was really able to fight with him on track but I'm looking forward to the day that happens."
The collision between Verstappen and Force India driver Ocon prompted an angry exchange between the pair after the race.
Verstappen remonstrated with the Frenchman, who is part of Mercedes' driver development programme, in the driver weighing area, pushing him three times.
Mercedes boss Toto Wolff was impressed with the restraint shown by Ocon during the confrontation and tipped him for major success in the future.
"Esteban is one of the future champions, that's for sure," said Wolff.
"He has shown some great performances this year. This guy is going to be on the very top spot on a podium in the future and can maybe one day be a world champion.
"When you look at where Lewis is today and how he has developed as a personality and as a driver, it is not only about driving a car fast and having talent.
"It is also about containing yourself, having the right social skills and intellect to progress and develop, and Esteban very much grows into this personality.
"We spend a lot of time talking to each other and developing him and I'm very happy the way it goes."
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