Here's my overview of personal favorites (& preferences):
A. Period versions:
Like others on this thread, William Christie's Erato recording of the Mozart Requiem is one of my 'go-to' CDs, and if I were pressed to pick a single favorite period version, I'd be strongly inclined to make it my first choice among period recordings of the Requiem (that I know). In a side by side comparison to Gardiner's well regarded Philips recording, for example, I find Christie's reading to be preferable by a good margin. He is arguably the finest Mozart conductor today. As for his edition, Christie used the Süssmayr completion:
https://www.amazon.com/Mozart-Panza...=1-1&keywords=william+christie+mozart+requiem
However, there are four other period Requiems that I enjoy equally or nearly as much (each for different reasons). They are:
(1) Christopher Hogwood's 1983 Requiem on L'Oiseau Lyre makes an excellent complimentary choice. He used the C.R.F. Maunder edition that sought to strip away all of Josef Von Eybler & Franz Xaver Süssmayr contributions and revise the orchestration according to Mozart's own sketches & style during his late period, while eliminating the Sanctus, Osanna, & Benedictus movements as not containing any genuine Mozart (though Robert Levin argues otherwise, claiming that there is some authentic Mozart in these movements). The Lacrimosa is largely reworked, with Maunder using material from the Introit. Most interestingly, he uses a rediscovered Mozart sketch for The Amen section of the Lacrimosa. Basically, Maunder takes the view that Süssmayr was no Mozart.
The problem with Maunder's completion is that we know Mozart was desperate to finish his Requiem in his last days, and worked feverishly to do so on his death bed, which means that we shouldn't rule out the possibility that when he became too ill to write anything down in his own hand, that he may have dictated parts of the Requiem to his pupil Süssmayr--such as parts of the Agnus Dei for instance. It is known that in his final days Mozart did leave detailed instructions to Sussmayr as to how the Requiem should be completed after his death. So, it's entirely possible that some last minute dictation took place between Mozart & his pupil that didn't end up in the existing sketches (or existed in fragments that Süssmayr later threw out, which were based upon Mozart's instructions). For example, it should be pointed out that nothing in Süssmayr's own church music prepares us for the high quality of his completion of the Agnus Dei. The other problem with getting rid of Süssmayr's contributions is that it leaves us with a Mass that is more noticeably unfinished and incomplete:
https://www.amazon.com/Mozart-Requi...1&keywords=christopher+hogwood+mozart+requiem
2. Nikolaus Harnoncourt's 2nd recording on hybrid SACD-DSD, with the Concentus Musicus Wien, is another period performance that I wouldn't want to be without. Harnoncourt's live performance uses the Beyer edition of the Süssmayr completion, though the Lacrimosa is quite different from the Süssmayr edition. Beethoven called the Requiem "a wild and terrible" work, and indeed Harnoncourt brings out the dramatic (& dynamic) extremes in the music better than most. In addition, it helps that the sound engineering is of a state of the art audiophile quality:
https://www.amazon.com/Mozart-SCHAF.../ref=tmm_acd_swatch_0?_encoding=UTF8&qid=&sr=
https://www.amazon.com/Mozart-Requi...FBWCPXFKPYB&psc=1&refRID=WKPYAMC1MFBWCPXFKPYB
3. Ton Koopman's live Erato recording is another intensely dramatic version of the Requiem. It gets my vote for the most underrated period recording of the work. While it's true that Koopman presents an "intimate, unfussy, unpretentious reading" according to Gramophone, there is nevertheless a 'hellfire and brimstone' quality to his reading that is impressive. That is partly due to Koopman's nimble chamber orchestra & choir (compared to 'big band' versions), as his smaller forces allow the listener to hear the entire score with an unusual degree of rhythmic clarity & transparency. The trumpets & drums, for instance, are more audible & thrilling in the Dies irae than in most other versions. As for the soloists, soprano Barbara Schlick is a favorite of mine in the "Recordare", & elsewhere. Koopman uses the Sussmayr edition, & Erato provides excellent sound quality.
https://www.amazon.com/Mozart-Requi...mr0&keywords=ton+Koopman+mozart+requiem+erato
https://www.amazon.com/Mozart-Requi...150450&sr=8-4&keywords=Koopman+mozart+requiem
4. I've also enjoyed the Mozart Requiem recording by the Collegium Cartusianum Köln, conducted by Peter Neumann, which can be bought individually, or in an inexpensive EMI box set, which IMO is preferable to Harnoncourt's Mozart sacred music box set on Teldec (though not as complete):
https://www.amazon.com/Complete-Mas...&qid=1545237310&sr=1-5&keywords=peter+neumann. In my view, Neumann is an underrated conductor in Mozart's sacred music, and his Requiem is one of the finest on record, IMO (& easy to underestimate). Neumann used the Süssmayr completion:
5. For those looking to understand a fuller range of ideas about how Mozart's Requiem should be completed, it's valuable to hear a recording that uses the renowned Mozart & Haydn scholar H.C. Robbins Landon edition of the Requiem, as well. Basically, Robbins Landon sought to retain Josef Von Eybler's abandoned first reconstructions of the earlier parts of the Requiem, in the belief that Eybler (who was Constanza Mozart's first choice to complete the Requiem before Süssmayr) was a more accomplished, inventive composer than Sussmayr, while also using Sussmayr's later contributions. (By the way, Masaaki Suzuki uses a newly commissioned edition that seeks to do the same, for his recent BIS recording, but it isn't Robbins Landon's edition.) However, I don't have a strong preference among recordings of the Robbins Landon edition. Bruno Weil and Tafelmusik were, I believe, the first to use this edition, and it's a good recording:
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6. It's also worth hearing a recording of scholar/musician Robert Levin's well regarded edition of the Requiem, too, but again, I'm not sure which recording is most ideal. However, Martin Pearlman & Boston Baroque's performance of the Levin edition is very good, on period instruments, while Sir Charles Mackerras & the Scottish Chamber Orchestra make an excellent choice on modern instruments:
https://www.amazon.com/Mozart-Requi...r=1-1&keywords=martin+pearlman+mozart+requiem
https://www.amazon.com/Mozart-Requi...9646&sr=1-1&keywords=Mackerras+mozart+requiem
I've not heard Frans Bruggen's live Requiem from Tokyo on the Glossa label, nor do I know what edition he used:
https://www.amazon.com/Mozart-Requiem-Bruggen/dp/B0035WARTE
http://www.arkivmusic.com/classical/album.jsp?album_id=35025
I don't know the Requiem recordings by Suzuki, Jacobs, Currentzis, or Butt's reconstruction of the premiere performance, either. Nor can I comment on Herreweghe's recording, since I haven't listened to it in ages (which may tell you something), though it has generally received favorable reviews.
B. Modern instrument versions:
Among versions of the Requiem performed on modern instruments, the 1980s Philips recording by conductor Peter Schreier & the Staatskapelle Dresden is excellent, and would be my first pick. The Staatskapelle is an orchestra that excels in Mozart, and the Leipzig Radio Choir are likewise very fine. Schreier used Sussmayr's completion, and his soloists are excellent, especially Margaret Price & Theo Adam. One of the aspects that I most like about Schreier's performance is that he exhibits a truer understanding of 18th century performing style (& practices) than the majority of non-HIP conductors on record, especially in comparison to the 'old school' conductors. It's a very beautiful performance, & well recorded by the Philips engineers:
https://www.amazon.com/Requiem-SCHMIDT-STAATSKAPELLE-DRESDEN-SCHREIER/dp/B00005IB5E
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Among HIP versions on modern instruments, I'd also recommend Sir Charles Mackerras' version with the Scottish Chamber Orchestra on Linn hybrid SACD (which, as mentioned, used the Levin edition). As with Schreier, I find Mackerras's version is preferable to the super slow 'old school' performances, which drag the music by performing the score in a misguided & wildly distorted post-Romantic style that Mozart wouldn't have recognized (or likely intended):
Finally, one of my biggest grievances with performances of Mozart's Requiem is that certain conductors take a substantial pause between the Kyrie and Dies irae movements. IMO, the latter movement works far better when the audience is led to believe that the Kyrie movement has ended, and then shockingly and most startlingly, the conductor and orchestra launch straight away into the Die irae, taking only the slightest pause between the two movements. It works much better that way in concert, at least from my experience. On the other hand, when a conductor takes a lengthy pause between the two movements, in order for the audience to fully cough out, and the musicians to get a breather in order to gather themselves before for the next movement, it only deflates the impact of opening of the Dies irae, IMO. Some conductors understand this, some don't (or if they do the record companies have separated the two movements with a pause between the CD tracks). Personally, I can't imagine that Mozart didn't want the two movements to be performed as virtually one continuous movement, with only a brief pause taken prior to the Dies irae. I expect he would have mischievously enjoyed seeing the audience think that the Kyrie was coming to an end, with the expectation that they could now sit back in their seats and relax before the next movement, & then bam!, the choir launches into the terrifying Dies irae. It's one of the most riveting, ingenious, & brilliant movements in music history, so it has to start right, IMO, and not come after the audience and musicians have taken a good long rest after the Kyrie ends. Here Maris Jansons conducts the two movements in continuity, the way that I believe Mozart imagined it (the Dies irae starts at 38:00, for anyone that wishes to go directly to the transition, though it's a fine performance... I expect you'll need to turn up the volume to better get the effect, as of course it's a lot more impactful in the concert hall than on a tiny You Tube screen):
. (Peter Neumann does the same, on the above YT clip.)
My five cents.