I am just wondering if supplying compressed hydrogen to a turbine would be a better fuel as it has three times the octane rating of kerosene. It just seems better all around.
Yes, indeed the plan to build a hydrogen-powered jet laid the groundwork for using hydrogen in the Centaur rockets and the upper stages of the Saturn 5. While hydrogen was used only experimentally in test rigs, the Soviet Union built a derivative of a regular airliner, the Tupolev 155, for testing hydrogen and natural gas in flight.
EDIT: NACA also did in-flight testing. Please see @jayhendren's answer for details.
So yes, the Tu-155 was indeed flown on several occasions with one hydrogen-powered turbofan. While the left and center engine remained NK-8s, the right engine was replaced by a NK-88 which was adapted for LNG and hydrogen. Other projects like one for a hydrogen-powered supersonic airliner sadly ended with the Soviet Union itself.
Tu-155 Cutaway view (picture source)
Hydrogen is also the prime propellant in hypersonic ramjets, but those are not turbines.
Hydrogen has a wide mixing ratio with air where it will burn. Also, being gaseous, it mixes much more quickly with air, so the combustion chamber can be small. When a converted J-57 was experimentally run on hydrogen in 1957,
The test engineers were agreeably surprised by the ease of engine operation. They ran it at full power and throttled back so far that the air fan was revolving so slowly the individual blades could be counted. Under this latter condition, the throttle could be opened and the engine would quickly and smoothly accelerate to full power. They found that the temperature distribution was good and there were no major problems.
But Niels is correct - its low density makes hydrogen problematic. As Alexis W. Lemmon, Jr., reported in May 1945 in his report on possible jet fuels (from history.nasa.gov):
"Although the liquid hydrogen-liquid oxygen system has by far the highest specific impulse performance of any system considered in this report, the low average density of the fuel components almost completely eliminates this system from all but very minor applications."
For fuels intended for use in aircraft, the key performance parameter is the energy density of the fuel: how much potential work is stored in how many liters of stuff. High energy density means the fuel tanks will be small and the energy released upon burning a liter of it will be large. The problem with using hydrogen as aircraft fuel is that its energy density is way lower than that of kerosene or diesel (because a liter of hydrogen at atmospheric pressure contains far less chemical potential energy than does a liter of kerosene), and squeezing it down to reduce its volume requires cryogenic refrigeration which hugely increases cost and weight, and the octane rating advantage of H2 is not balanced by these disadvantages.
Specifically regarding burning hydrogen in brayton cycle turbines, this is possible but economically impractical because the cost to make a liter of hydrogen is far greater than the cost to refine a liter of jet kerosene from crude oil.
Von Ohain's first prototype of his HeS 3 turbojet, the HeS 1, burned hydrogen in the first runs. Only after some modifications was he able to make it work with a liquid fuel.
This is not fully on topic ('Aviation'), but answers the question in some way:
Hydrogen is currently considered the 'fuel of the future' for existing and new gas turbines in power plants, for small and large gas turbines equally. The industry is working to enable all existing turbine lines to handle that, and the modifications are minor (well, minor compared to the complexity of a modern gas turbine). Many models are already successfully enabled and proven, and the market is expecting to get the first requests soon.
Overall, it is quite easy to use hydrogen - just maybe not on an airplane.
The main reason to go for hydrogen burning is that it enables hydrogen storage as a battery - when there is extra power available (preferably solar or wind), it gets converted to hydrogen, and when the power is needed, it gets burned in existing gas turbines. This would allow to save trillions of existing investments in power plants by converting them into zero-emission hydrogen-burning plants.
[Disclaimer: I have a professional relationship to such a company; however, this is not restricted information]
NACA (the predecessor to NASA), has burned hydrogen in a turbojet engine in flight.
In the 1950s NACA burned hydrogen in a turbojet engine on a modified B-57 aircraft as part of Project Bee:
A technical report on the in-flight performance of the hydrogen-powered turbine was published and is available for download on the NASA website.
The experiments with hydrogen as a fuel source for turbine engines were successful enough that the Lockheed CL-400 high-altitude reconnaissance aircraft was designed to use hydrogen fuel:
Ultimately, however, the CL-400 was cancelled, although some of the research for this project made its way into the now-famous SR-71.