Your HR will almost certainly be lower across the board on the bike. You are using fewer muscles, etc. My Z2 on the run is 155, but 145 on the bike...

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Almost always different, usually bike is lower.

However let me say this, you'll be better off using a threshold heart rate estimate to set your training zones rather than max heart rate. Easier to test for one and the test itself is a training session.

You can google lthr and joe friel and use the 30 minute time trial he uses (you go all out for 30 minutes and then use the averae hr over the last 20 of that 30 minutes all out). The if your threshold heart rate goes up or down, your test will reflect it.

And yes, you need a separate test for bike and for run.

Most of the time threshold heart time on the bike in 10bpm lower than on the run

And yes you need to perform two tests to get your training zones. One for the run and one for the bike.

All true, but they certainly do not add together to saying that cycling uses less oxygen and therefore needs less HR. It is just that most of us are better at pushing harder on feet than on wheels. Most of the strong cyclists have power and therefore don't add their 2 cents.

So yes, definitely test them separately. But just because your cycling LTHR is 5, 10 or 15 bpm lower than your running number, don't assume that if you retest one in 6 months that you can apply that addition to know the other.

Actually if I recall correctly you do actually use more oxygen (like effort for like effort) on the run than bike. The limiter on endurance performance at a physiological level is not how fast your heart can beat or how hard you can push yourself on bike vs run, it's much more closely related to what goes on at the mitochondrial level within the muscles. I'm not sure on the specifics but I think I remember reading that somewhere?

And yes I do have power and my LT HR on the bike is about 8-10bpm lower than my run, and I've pushed hard enough to make myself vomit on both the bike and run

Yes.

Have some fun and do the run and bike field LT tests to find out your HR averages, read this on field LT testing protocol.

Then use the downloadable spreadsheet here for determining the HR zone calculations using the above field LT test HR average results.

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I have found that as I get very fit my max HR is very similar on Bike and Run, when I started biking after coming from a run back ground there was a much bigger gap, I am in my late 40's so not sure this makes much difference.

This is not the best paper about it, but all I could scratch up on the spot: http://jap.physiology.org/content/16/6/977 The abstract concludes: "It is concluded that the aerobic capacity and maximal heart rate are the same in maximal running or cycling, at least in well-trained subjects."

Perhaps later, I can find the better citations. Basically, they all read that yeah, running uses more muscle groups, but you end up using those few leg muscles in cycling much harder. So in the end, you're cycling at the same maximal level as running...which is simply your max possible. If your limiter is HR, then that pegs first. If its mitochondrial activity, then I'd assume those leg muscles in well trained cyclists get more dense to even out first.