Apart from the code, wich you can look though of course, here the mechanics of that guy:

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This AI is based on finding a movement path without
the need for creating a node-based pathfinding grid.

The nodes are created "on the fly" by the player when he moves.

The player continiusly "drops" nodes when walking.
As the nodes are in a linked list, they somehow form a
"chain", wich is the path for the AI to move.

The AI guy looks for the most recent node, that he can directly reach.
Once he is there, he will go to an even more recent node.
Similar to a Dog, sniffing his way.

This way the AI follows the player.

I dont remember exactly, but I think, when the AI gets stuck,
he walks back his way, until he can find a suitable node again
to follow.

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The other part is the movement script, wich uses
several traces, that makes the AI avoid to get to close to walls.
So before colliding directly with something, the AI usually pushed itself away from the obstacle or turned a bit.

The same with jumping. The Players actually hover over the ground up to the hips.
Similar to humans, the legs are basically
fully flexible to adapt to narrow obstacles on the floor.

If the trace dow finds a distance lower than the leg-hight, the
AI pushes itself up. If it is higher than the hip-hight, it
stops pushing, and starts to fall.

Thank you Damocles! =) This is very usefull!!!

when player jumps , companion moves under him keeping player on top......

I think thats becouse of player.x, "companion" is trying to get in to player.x position

The companion tries to get to the player as close as possible,
and both player and bot automatically step onto obstacles to move over them.

Here the bot sees the player as objstacle and steps onto him, or
the other way around.

you can avoid that by letting the bot stop earlier or let the bot not step onto the player (ignoring him in the trace)

I'll make if comparison, with distanse to player, if closer then 400 quants, follow player, if closer 50 quants stop & attack... I think that way this AI is the best for my zombie shooter game!

Does the companion attack enemies, if they attack the player?

No thats not implemented in that code.

Its basically just a demonstration to get
an active entity to follow you.

To extend it to an attacking companion,
you need to let the companion switch from
following the player to chasing the enemy.

(using a statemachine or something similar)

OK, thanks for the help