Purple Time Rift
It's lacking critical information, as well as foundational structures.
A Purple Time Rift (known also as Memory Rifts) are rifts where they explore the past memories of the residences through story-driven sections, currently there's up to 10 Purple Time Rifts.
About
Unlike other levels and Blue Time Rifts, a Purple Time Rift lacks checkpoints and must be completed in one go, the rift is separated into sections known as floors.
Each floor can be completed by collecting the required amount of 
Rift Pons to be spent to unlock the Rift Gate that leads to the next floor. You keep leftover Rift Pons from previous floors, even to a point you can skip multiple floors at once if you collected enough.
The final floor contains no Rift Pons, however, a Purple Time Rift Ball is contained in it, the only way to progress further is by striking the Rift Ball 4 times for it to break, giving an abundance of 
Pons and when shattered a 
Time Piece appears to be collected.
Additionally, there are 
Storybook Pages scattered around a Purple Time Rift that can collected, upon collecting the Time Piece related to the level, the storybooks merge together to create a Storybook.
Unlocking a Purple Rift
To unlock a Purple Rift you must collect relics related to each chapter, order them correctly onto the Relic Stand.
When a Relic Stand is completed, a Purple Rift Ball appears orbiting around a telescope for a chapter, telling you of a Purple Time Rift that was unlocked.
List of Purple Time Rifts
Base Game
A Hat in Time comes with 4 Purple Time Rifts, each can be unlocked by completing a relevant Relic to its chapter.
Seal the Deal
In Seal the Deal, 5 Purple Time Rift were introduced. 1 through completing a Relic related to the The Arctic Cruise chapter. The other 3 are Community Time Rifts that can be found through Death Wish, these rifts are slightly difficult due being considered a Death Wish contract, resulting in activating the Death Wish Difficulty modifiers[citation needed].
The last Purple Time Rift, Tour, is unlocked by getting 
x45 (x55 with Nyakuza Metro installed) Time Pieces to appear, Tour is designed to be the last Time Piece you gain to achieve 100% completionist when it comes to Time Pieces collected.
Nyakuza Metro
In Nyakuza Metro, 1 Purple Time Rift was introduced which is unlocked by completing a Relic related to the Nyakuza Metro chapter.