In gem mineralogy, Peridots and Amethysts go well together. This also formed Rupphire, because Rubys and Sapphires went well together as well.
In "Jail Break", during Garnet's song, Amethyst ties up Peridot in her whip.
In "Catch and Release", Peridot gets discovered by the crystal gems when she was un-poofed by Steven. She is seen running up the stairs, and Amethyst is sitting on Steven's bed, her legs crossed while she was pulling her whip suggestively. This part was in Peridot's perspective of Amethyst, and Amethyst seemed to look rather attractive and seductive in the scene.
In the beginning of "Too Far", Peridot describes multiple objects in unconventional ways, amusing Amethyst greatly. She then deems Peridot "funny" and as a result she tries to impress Amethyst and make her laugh.
In "Too Far", Peridot accidentally starts the drill and she falls off it, sending it wildly spinning towards an unsuspecting Amethyst. Peridot tries to get up and shouts for Amethyst with genuine concern. This may be the first time Peridot put someone's well being before hers.
The most glaring hint is when Peridot accidentally mounts Amethyst after pushing her away from the Drill. They stare at each other for some time before Peridot anxiously jumps up and claims it was an accident.
In the same episode, Peridot unknowingly offends Amethyst, causing Amethyst to stop giving her attention. Peridot is very confused about what she's feeling, Steven explains that she feels guilty for hurting Amethyst's feelings. She is later seen apologizing to Amethyst through her voice recorder, thumbs nervously twitching as Amethyst listens. After having her apology accepted, she genuinely smiles and says she feels "big".
In "Steven's Birthday" Amethyst blows up a Peridot shaped balloon and pops it in front of Peridot.
In "It Could've Been Great" Peridot seems offended that Amethyst brushed off her knowledge on how Gem's automatically adjust to gravity.
In "It Could've Been Great" Peridot compliments Amethyst by saying the Kindergarten near the cluster was not nearly as impressive as Amethyst's.
In "Message Received" Amethyst, as well as Steven, seems the most hurt about Peridot's betrayal, in such she revokes all her nicknames for her.
In "Log Date 7 15 2" Peridot blushes after Amethyst says "Pretty impressive in my opinion but it's hard to beat the original" after shapeshifting into Peridot's form.
In "Log Date 7 15 2" some people suspect that the Peridot's shipping chart was to resemble the gems. The green box resembling Peridot and the purple Amethyst have a squiggly line connecting each other.
During Beta, Peridot gets super excited to see Amethyst and ignores the fact that Steven is there.
During Beta, Peridot tries to impress Amethyst several times and admits to it.
During Beta and Too Short to Ride, the two seem to be comfortable with touching each other.
Peridot was overprotective over Amethyst at the end of the episode. Convincing her not to fight Jasper alone (or at least trying to). She knew she was going to get hurt some how, poofed or shattered
Both Amethyst and Peridot are classified as defected; Amethyst incubating for too long and coming out short, and Peridot being an "Era 2 Peridot" who is unable to do most standard gem abilities.
Rebecca Sugar has said that pushing someone out of harm’s way "an outrageously romantic gesture". Leaning towards Amedot, Stevonnie, and Rupphire.
In "Steven's Birthday", Amethyst blows up a Peridot shaped balloon and pops it in front of Peridot.
In "It Could've Been Great", Peridot seems offended that Amethyst brushed off her knowledge on how Gem's automatically adjust to gravity.
In the same episode, Peridot compliments Amethyst by saying the Kindergarten near the cluster was not nearly as impressive as Amethyst's.
In "Message Received", Amethyst seems the most hurt about Peridot's betrayal, in such she revokes all her nicknames for her.
Amethyst gives Peridot's more nicknames then she's given to anyone. Ex. Pear, Peridactyl, Peri, etc.