The three-time champion said he welcomed the idea that Formula One drivers should race in other series.
Speaking ahead of this weekend’s Bahrain Grand Prix, where the Mercedes driver expects a very close race in torrid heat against Ferrari's Sebastian Vettel, the three-time champion said he welcomed the idea that Formula One drivers should race in other series.
"I think it's great that a driver is able to do that," Hamilton said of two-time champion Spaniard Alonso's switch to the Indianapolis 500, instead of the Monaco Grand Prix, next month.
"I think us drivers should be able to do more than one series. There was a period of time in the past where there were drivers doing multiple series. I think it's pretty cool that he's doing it."
Alonso will race at Indianapolis for a team entered by McLaren, Honda and Andretti Autosport.
Hamilton added: "I wouldn't miss any of the races in F1 – I’d definitely continue to do all the races. I'd like to do MotoGP. I'd like to ride a MotoGP race and probably a NASCAR race like the Daytona 500, maybe…."
Briton Button remains contracted to McLaren after making way for Belgian Stoffel Vandoorne to partner Alonso this season.
He is the obvious leading contender for the seat, but the team has yet to make any announcement.
"I hope Jenson comes back,” said Hamilton of his former team-mate. "It would be great for the sport to have Jenson back.
"I still think he is one of the best drivers. I like him – for his calibre as a driver and for his great experience."
As to this weekend’s race in temperatures of up to 40 degrees Celsius, Hamilton said he is expecting a tough contest.
"We are working so hard as a team to start this season well," he said. “But Ferrari are too and it is very close.
"They are a team that has great pace, maybe better pace, and they are always very quick on a hot circuit… I am sure the gap is going to be very much closer.”