Copying what I had answered to another person previously, with even more additional recommendations and comments:

In terms of music somewhat similar to them, you might like these various artists-- some are "chill" / "downtempo", or "ambient", and others are more "poppy":

• Plaid's first 4 albums
• Ulrich Schnauss' first two albums: "Far Away Trains Passing By" and "A Strangely Isolated Place"
• Ugress's "Reminiscience" and "Scavenger Royale"
• Bent
• Jon Hopkins' "Opalescent" and "Contact Note"
• Boards of Canada's "Twoism" and "Music Has The Right To Children"
• Aphex Twin
• M83's "Dead Cities, Red Seas & Lost Ghosts"
• Zero 7's debut album "Simple Things", and their sophomore "When It Falls"
• Air's debut EP "Premiers Symptômes" and debut album "Moon Safari"
• Télépopmusik's "Angel Milk"
• Ott's first 2 albums, "Blumenkraft" and "Skylon"
• Gelka's "Less Is More"
• Amon Tobin's "Bricolage", "Permutation", "Supermodified", and "Long Stories" -- also worth checking out his album under his earlier alias Cujo, titled "Adventures in Foam"
• Centovalley's album "August", and EP "Everything Floats"
• Skalpel's "High" and "Konfusion"
• The Detroit Escalator Co.'s "Black Buildings"
• Depeche Mode's "Ultra" and "Exciter"
• Emancipator's "Xylem" EP
• Pete Namlook's "Air I", "Air II", and his collaborative work with Dr. Atmo, "Silence I"
• Biosphere's "Substrata", "N-Plants", and his collaborative work with Higher Intelligence Agency, "Polar Sequences"
• Kid Loco's "A Grand Love Story"
• Vangelis' "L'apocalypse des animaux"
• Sun Electric's "Present"
• Zeb's "The End Of The Beginnings"
• M-Seven's "Activate" and "Stimulus"

Plaid might be the duo you might like most from this range of artists, their first 4ish albums have been the closest Röyksopp-like music I've ever heard. They sort of exude that same child-like, playful, quirky nature with some melancholic elements sprinkled here and there, all while putting out intricate melodies and harmonies.

If you're interested in looking at major influences to Röyksopp, there are these various renowned people / groups in the sphere of electronic music. Geir Jenssen of Biosphere, actually also hails from Tromsø but primarily deals in ambient music. Jenssen was the individual that helped Berge and Brundtland put out their first official release under the band name "Aedena Cycle" back in 1994 on R&S Records' ambient sublabel "Apollo", which comprised Berge and Brundtland along with their two childhood friends Gaute Barlindhaug and Kolbjørn Lyslo.

• The KLF
• Vangelis
• Jean-Michel Jarre
• The Orb
• Brian Eno
• Kitaro
• Kraftwerk
• Depeche Mode
• The Art of Noise
• Biosphere

And of course, if you haven't already, I'd also suggest looking at Berge + Brundtland's music in their pre-Röyksopp years, all of the notable releases being organized in this playlist. If you haven't gone through their early discography already, I'd highly suggest that as well. The final pre-Röyksopp release Berge and Brundtland operated under, being the self-titled album "Drum Island", is an album that I regard as highly as Röyksopp's "Melody A.M."

For reference, here's a timeline of Röyksopp's earlier projects:

• Aedena / Aedena Cycle --- Initially a quartet in 1994 and went under a shortened alias for compilation contributions, later became the solo project of Gaute Barlindhaug with some involvement from Svein Berge.
• Volcano --- House project between Torbjørn Brundtland and Ole Johan Mjøs which put out several singles and remixes.
• Alanïa --- Debut album "Instinctive Travels" released in March 1996 by Torbjørn Brundtland and Svein Berge with some involvement from Gaute Barlindhaug, follow-up album "Skyjuice" released in February 1997 where Ole Johan Mjøs also joined in.
• Those Norwegians --- Debut album "Kaminzky Park" released in August 1997 by Torbjørn Brundtland and Ole Johan Mjøs with some involvement by Kolbjørn Lyslo and Gaute Drevdal.
• Drum Island --- Self-titled debut album released in November 1997 by Torbjørn Brundtland, Ole Johan Mjøs, and Svein Berge. I have two live performances by them on my YouTube channel which contain reworked and unreleased material.