Lode Runner

from Lode Runner English box blurb

You're a highly trained Galactic Commando, deep in enemy territory. Your mission: to recover a fortune in gold, stolen from the Bungeling people by their power hungry leaders.
You'll be running, jumping and climbing heroically, solving perplexing puzzles. With your laser pistol, you'll drill passageways through stone floors and barriers. To get through this mission alive, you'll need more than fleet feet and good looks. You'll need your quick wits and brains.
And more than just a little luck.

Run around, climb ladders, collect treasures, make pits to drop down to lower floors or trap enemies, and don't caught by the robots. Despite the English blurb's statement, no actual jumping takes place. (how does one climb heroically for that matter? There are a number of fine online resources for Lode Runner information, including Lode Runner Forever!, the Lode Runner Museum, the Lode Runner Archive, and, of course, the good ol' Wikipedia article. That's all fine and well, but you might be wondering...

Why is Lode Runner on a Bomberman site?

I'm glad you asked!

Originally developed by Douglas E. Smith and published on English shores by Broderbund, Hudson got their paws on the license to produce their own versions of the game (sees the notes section for links to the relevant pages), and it just so happens that before Bomberman got his own game, he was co-starring in this title as the enemies that constantly pursue the gold hoarding hero.

Lode Runner is the second in an unofficial trilogy of games by Broderbund, the first and third being Raid on Bungeling Bay and Choplifter respectively; all of which feature the war on the evil Bungeling empire. The original English computer versions used simplistic stickmen figures to represent the player character, and the only other representation of the robots was on the stylised box art, where they appeared like typical robots from pulp science fiction.

Hudson opted for their own vision, rendering the characters in a cutesy cartoon manner, though the look varies between gameplay and the box art. Bomberman apparently acts as a prequel to Lode Runner, detailing the plight of one of the Bungeling empire's slave labourers, who proceeds to escape and become human.
I have no idea if Bomberman's NES game was planned while Hudson ported Lode Runner, but all I know is that given the new designs are Hudson's visual creation, they could get away with making a spin-off of another company's product without reprimand. How else would we have nearly 30 years of Bomberman games otherwise?

Needless to say, this completely complicates the story, given how Bomberman is a spin-off that ties back into the instalment it spawned from, and can really complicate the matter of what the hell Bomberman is - is he a true robot, despite being turned into a human for him to become the hero of Lode Runner? Is he a cyborg that was merely stripped of his robotic implants?
Given how the Bungeling trilogy never uses the robots again, I can't say it matters much. The Bungeling empire is never referred to in any other Bomberman games, so you could argue that the NES Bomberman game is non-canon.
Aw yeah, I went there.

This wasn't the only Lode Runner game to have connections to Bomberman. Check out the pages for Battle Lode Runner and Cubic Lode Runner.