What brackets do I need to anchor a new post over an existing wooden post ?

Can you replace the post, concrete and all? If you don't, and try to splice on a 4"x4", it's not going to be too stable.

Someone else is going to come along and help you with a quick fix.

1) Consider drilling down into the hole with a ,let's say, 1/2" bit,as far as you can reach with drill and bit,then thread the longest lag bolt down the new hole and try to remove the remaining post.

2) Being a gate post it will get more use/pressures than even the hinge side,the post needs to be secured as best as you possibly can.(as jmj said)Without going to a descent depth,you are just spinning your wheels,you'll be replacing this one again in the near future.

Removal and replacement of the concrete base with post buried might be a little labor,but it's better to do it right once,than wrong once and then right on the second try.

Everyone told you what to do: replace the post
That's a nobrainer.

No one told you how:
Rent a demolition hammer and chisels

To keep that from happening again, coat the bottom and sides of the NEW post that's being buried in the Crete with 2 coats of foundation coating before setting it back in concrete to STOP this rotting.

All to common a problem. PT. Has its limits

To really tweak it slope the Crete away from the buried post so ground water doesn't just sit on that flat ledge. After set, use foundation coating for the rest of the buried post.

Again PT has it limits

I wouldn't rent a jack hammer UNTIL I tried to dig a hole next to the concrete "chunk", and try to get it out in one big piece. Break it up with a sledge hammer if needed.

It's probably not built to military specifications. Just dig,dig,dig.

By enlarging the hole you'll compromise the native compacted soil, so for replacement purposes minimal disturb of the surrounding soil is necessary.

Your also missing the "leverage" component that you would have if you were removing say a 6' post anchored with cement.

Remember that Home depot rents tools.

Sorry but no quick fix here. What you had took years to fail and the remaining concrete is just not worth trying to salvage. I'd yank it out with as little digging as possible. When replacing the post use a PT rated for below grade, most of the stuff you see in box stores is not. Also put several inches of gravel in the bottom of the hole, set you post, another few inches of gravel before the new concrete. This stops the concrete from making a concrete bowl that holds water. Last but not least make sure your hole is slightly larger at the bottom and the slowly narrows as it comes up to grade. This should stop frost in the gound from being able to slowly lift the concrete plug over time.