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The article entitled "54 Celsius degrees in Iran, the highest temperature ever recorded?" says 56,6 Celsius may be the highest temperature ever recorded, in Death Valley, California, and it also mentions that Iran's heat index peaked at 61,2 Celsius in the event the story was about.

I ask this question because several cities in my province in Argentina are registering heat indexes of over 60 Celsius, with a maximum in the center of Santa Fe city of 65,7° Celsius.

So what was the highest temperature ever recorded on the surface of the Earth?

The record the article is referring about seems to be the same as registered at Guiness World Records:

On 13 September 2012 the World Meteorological Organisation
disqualified the record for the highest recorded temperature, exactly
90 years after it had been established at El Azizia, Libya, with a
measurement of 58°C. The official highest recorded temperature is now
56.7°C (134°F), which was measured on 10 July 1913 at Greenland Ranch, Death Valley, California, USA.

Those records are usually well researched, and in this case it matches what is reported by the WMO. Since 2007, the WMO Commission for Climatology (CCl) has maintained the Archive of World Weather and Climate Extremes. In that archive, the same record is reported as the current highest temperature ever recorded:

More details about the record are described here.
The official WMO press release about the invalidation of the 1922 El Azizia record can be found here.

It is surprising however, that with most the hottest years on record happening in the last few decades, the maximum recorded temperature is still from 1913.

Some people have questioned the reliability of those old measurements. More recent and much more reliable measurements seem to top-up at about 54.0°C as pointed out by a very interesting article brought up by @Plutor in the comments.

Note that you are mixing temperature and heat index in your question. Make sure you understand the difference.

PS: "Greenland" Ranch... a rather sarcastic name for such a place.

Photograph of old Greenland Ranch Station, California, USA (source)

The Hadean era covers the time from the formation of the Earth until 4bn years ago. It was characterized by a surface of molten rock, due to repeated meteor strikes, volcanism and radioactive decay.

"Liquid water oceans existed despite the surface temperature of 230 °C (446 °F) because of the atmospheric pressure of the heavy carbon dioxide atmosphere."

Whether this counts as "recorded" is another matter. No-one measured it with a thermometer, but it can be modelled using palaeo-climatology principles.