Nice tip Nick, good to know, and easy to check for.

Ta-da--------- It was the flasher units. 

New flasher, signals working.   Relief.   This was a bare shell build I did 25 years ago. I took it all apart to get it repainted and now doing final assembly. I feel like I'm building a prototype.  I ended up with 2 NEW flashers that were no good. 3rd one is the charm.

It appears to be the flasher unit. I got a new one today with a open box. When I installed it, it lit the signals but would not flash. And it did not light the stalk in the neutral position.  Back to NAPA in the morning.  I'm glad that this is what it appears to be. Easy fix but PIA back and forth to the auto parts store.

Doug, I'll check that out. I had two new clear  housing flashers that I can't find. Too much stuff in my life.

Mike, +1 for the information above. 

I have seen this before.  In my case it was a wiring mistake at the flasher.  If you want to try something quick and don't mind spending a few dollars, get a Tridon EL13 electronic flasher from the parts store and wire it as shown in the picture on the link below:

Address:  https://sites.google.com/site/purlawson/home/pictures
Look for picture "AftermarketFlasher.JPG"

With the stalk in the neutral position use your voltmeter to check voltage at terminal P on the flasher unit. It should be zero but I suspect it must be 12V which says that there is an internal fault in the FU. The repro ones seem to be fairy crap so try to find a decent used one. 

I can see how the bulb might stay lit in the neutral position: wires are not correct on the flasher or the flasher is not correct.

Keep in mind that the flasher sends power to the indicator warning lamp, NOT the turn signals switch, as with later models.

Thanks Mur,    I've got the relay correct, and just looked at my notes. Yes I have the lgp wire feeding the signal stalk bulb.

I double checked the connections to the flasher and they are correct. Maybe take another look in the morning. I can't see how the bulb would stay lit in the neutral position.

So I guess insulating the clamp on the column is not my issue.

The green bulb on the end of the stalk is fed by the L.G.P. Wire that comes from the flasher switch.

L.G.N. Comes from the flasher to power either the left or right turn signals, when they are selected by the switch.

G. Comes from the fuse block terminal #4 and powers the flasher.

these wires are Green, Light Green with Brown stripe and Light Green with Purple Stripe.

Did you figure out that relay?

I've installed a new aftermarket mk 1 turn signal stalk in my 1960 restoration. It's got a new harness.  All the electrics are complete and working except this:  when I turn on the ignition, the green light on the end of the stalk lights.  If I put the stalk in right or left turn positions/ up-down, it blinks, but in the neutral position, the green lamp on the end of the stalk is still lit. I know this is not correct and it should be off in the neutral positon.

I've traced all the wires, and see that the green bulb on the stalk is fed from the fuse box position 4.  I've checked the continuity from one of the 3 green wires on #4 and found the feed.  #4 is only powered when the key is on.

While typing this I might have solved this but won't know until I try this:

is it possible that the stalk clamp to the column needs to be insulated?

All feedback is welcome.