No, this isn’t Jon and Damian having a hissy fit. It’s actually Clark Kent Jr. (aka Superman Jr. or Superman ll) and Bruce Wayne Jr. (aka Batman Jr. or Batman ll) in their first appearance as the Super-Sons, at least as a semi-regular feature in World’s Finest. There had been plenty of “imaginary stories” involving offspring of both Superman and Batman before this, but those stories were generally one-shots, and the children were much younger.
In the Super-Sons story series Superman Jr. and Batman Jr. (writer Bob Haney thankfully dropped the “ll” from their names after this initial appearance) were portrayed as typical rebellious teens, tired of living in their fathers’ shadows.
The sons looked almost identical to their fathers, especially in costume, although artist Dick Dillin would occasionally draw Superman Jr with mutton chop sideburns (which were popular at the time (1973).
The boys’ mothers were never identified, and when they appeared their faces were always drawn in shadows so deep it looked like their heads had been dunked in India ink. Superman’s wife was shown with black hair (leading everyone to assume she was Lois Lane). Whoever she was, she was definitely a normal human, because we’re told repeatedly that Superman Jr. half-Kryptonian/half-human.
Batman’s wife was usually drawn with red or auburn hair (which meant she could have been anyone, including Vicki Vale, Pamela Isley/Poison Ivy, Barbara Gordon/Batgirl).
Being half-Kryptonian, Superman Jr.’s power levels are only half of his father’s. In this first story, he is shown being unable to fly, but he can leap up to 100 miles in a single bound. Bullets don’t hurt him, but they leave nasty bruises (a trait he shared with Marvel’s Luke Cage). And while bombs don’t kill him, they render him unconscious. A hallmark of Bob Haney’s writing was lack of consistency, and Superman Jr. was no exception; by his second appearance he was flying effortlessly, although a strong headwind could slow him down.
The Super-Sons proved popular enough that thy appeared ten more times in World’s Finest over the course of the next three years. Writer Haney treated the stories as actual tales from the future and, in his mind at least, it was canon that Superman and Batman would have namesake sons who would embark on these adventures.
Haney’s final Super-Sons tale was in World’s Finest #242 (Dec. 1976). Not too long after that he was no longer the regular writer on the title, and by the end of the decade had pretty much stopped working for DC.
The Super-sons made one final appearance in World’s Finest in issue #263 (July, 1980), written by Denny O’Neil. The boys showed up in the adventure, much to the surprise of their “fathers.” It was revealed that the sons had never existed; they had been simulations Superman had run on his computer in the Fortress of Solitude to show Batman and him what it might be like if they had children. A bizarre accident had caused the computer simulations to come to life, but an energy they gave off was threatening to destroy the planet. Convinced that they weren’t real and posed a threat, the boys sacrificed themselves by diving into Superman’s disintegration pit* and saved the world.
This version of the Super-Sons has returned a few times since, either in Elseworlds tales or stories depicting different Earths in the multiverse. The last time I checked, versions of these Super-Sons were living on Earth-16, but they may have, or probably did, change with the introduction of the DC Omniverse.
* Superman described the disintergration pit, a regular feature in the Fortress at the time, thus: “It’s a sort of atomic cauldron, really, fueled by radioactive Kryptonian elements I obtained from the bottle-city of Kandor long ago. It’ll dissolve almost anything which is dropped into it back into its molecular components within an instant. And that would include yours truly, if I were feeling suicidal.”)