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Measuring temperature in joules instead in the artificial units of Kelvin would render entropy as a dimensionless quantity. This is quite appealing since entropy has always been quite a misterious quantity: it is used a measure of the disorder in a system but its units are J/K, which makes it really hard to interpret. The reason why entropy and temperature were defined this way is because they were studied before the athomical composition of matter was completeley accepted. You can read more about this in these posts:

Why isn't temperature measured in Joules? Should entropy have units and temperature in terms of energy?

Arieh-Ben Naim, a chemistry from Israel, has really good books in this topic such "Entropy demystified". He is one the main defenders that entropy should be a unitless quantity.

I have always like to understand the gist of things, and the first step to understand a magnitud like entropy is understanding how it can be measured. For instance, I think I know what a speed is because I can reason in terms of space and time. But in entropy, as the way is normally defined, this is hard and tricky, and all the explanations I have heard so far seem to me like quite far-fetched. So here it goes my question. Let's accept for the sake of argument that we are considering entropy a dimensionless quantity. Now we heat a gas, and we calculate the entropy associate to this process. The result would be a number, whatever. Let us say that number is going to be 100. How to interpret this number? What would it be measuring? And finally, do you agree with this vision of temperature units?

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Adrián A.D.
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What it means is that the number of bits required to specify the exact physical state the system is in, increases by 100/log(2) bits after the gas is heated.

I think measuring the temperature in energy units is a step in the right direction, but what is even better is to do without any units. I.e. while units may be introduced for convenience, the formalism we use in physics should not have units hard wired in them, as that's misleading and leads to stupidities as pointed out by Micheal Duff in this article.

Count Iblis
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I would strongly disagree with your statement that entropy has always been quite a mysterious quantity. Quite the contrary.

What makes water (without something else interfering) always flow down the hill? Gravity.

One does not have to know anything further about gravity to make good use of this statement, and people have used this fact for thousands of years without any further elaboration on the microscopic theories of gravity. Now ask yourself a similar question:

What makes heat (without something else interfering) always flow from the warmer to the colder temperature bath? Entropy.

Logically it's the same statement. We OBSERVE a process that has a preferred direction, and we call the cause of this observation by a name. In one case we call it "gravity", in the other we call it "entropy".

There is absolutely nothing "mystical" about entropy, once you know what it does and why one needs it (because there is not one observation of heat flowing from cold to hot without some other process facilitating it).

To use the same units for temperature as for energy is completely nonsensical, since the definition of energy is "the ability to perform work". Temperature, by itself, can not perform work, no matter how you look at it.

CuriousOne
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