Portal Rift by Laurie Woodward

Ever since Bartholomew Borax III returned to being homeschooled, Mother has been so cleaning-obsessed and cruel that all he dreamed of was finding another portal to the magical Artania.

Until now.

With doorways opening without warning and thrusting him through time and space, he has no idea where he'll end up next. It might be a Parisian loft with a depressed Monet, near a burning café with hunchbacked monsters in pursuit, or in the middle of an empty street at midnight.

And then, the real trouble begins - his best friend falls into a coma that no one can wake him from. Everyone is perplexed but there are clues; anagrams which Bartholomew must decipher. But with Alex unconscious and time running out, will Bartholomew find the key in time?