This is from Heinrich Franz Biber's Mystery Sonata 15:
The key signature is odd in that the F# appears in both octaves and the C# appears in its usual place, but an octave below it's natural. Does this actually mean that the C is sharp or not depending on the octave? If so, the piece is in 2 keys at once (and possibly breaks the "rule of the octave").
The "key signature" reflects scordatura tuning instructions — in which a stringed instrument is tuned in a way other than its standard — and the Biber Sonatas are mentioned in this regard in the strad.com article Peg-turning masterpieces: Heinrich Biber Baroque Scordatura
Rather than the usual G-D-A-E tuning, the notation before the staff indicates a tuning of G-C-G-D. The music itself is notated at "fingered pitch". So, for example, a notated open E string would be played open but would actually produce a D. In other words, the upper three strings in Biber's scordatura would all sound a whole step lower than the written pitch.
Below is the first few measures of Biber 15: first in scordatura (original), then transposed for standard tuning. Notice, for example, that the first note, written as A in the original, is written as a G in the standard-tuning version.
Now consider the second measure, beat 3. In the original, there is a D-C (natural) double-stop, but in the transposed version, it's just a C. That's because the notated D in the original will sound as a C — so both strings are playing the same pitch.
Similarly, the first note in measure 3 would be played as an F# in the original, but sound as an E, as notated in the transposition.
Then there's measure 4, note 3. The scordatura indicates to play a C# (as indicated in the "key signature"), but this will actually sound as a B.
One way to think about this specific case: The bottom string is playing in the key of C, but due to the scordatura, the upper strings, in order to sound in the key of C, must play in the key of D. That's why the accidentals have to be specified independently for each line or space affected: depending on the (re-)tuning, a particular pitch may be more often encountered as a transposed pitch than a "normal" pitch, and the transposition will depend on which string most frequently encounters that pitch.
Biber's "Rosary" or "Mystery Sonatas" make prominent use of scordatura; that is, they call for the violinist to tune the open strings to pitches other than the usual G, D, A, and E. (Except for the first and last sonata.)
The pitches you see in the violin part are not the pitches that would be produced. The stemless noteheads before the start of the staff indicate that the top string, the "E string," be tuned down to a D; the A string down to a G, and the D string down to a C. (The G string remains a G.) Then the notation that follows represents the fingering that the violinist uses; an A above middle C would normally be played by the open A string, but that now sounds as a G. The following B would be played by the first finger a step above, but it now sounds as an A; etc. Here are these opening two measures, notated at sounding pitch:
(That comes from this file but beware, apparently there are a lot of mistakes in Sonata XI in it. More sources, including manuscript, at IMDB.)
So the strange key signature helps the violinist put their fingers in the right places. The C#, for instance, is normally a major 3rd above the A string, and in this tuning ensures that we get a major 3rd above what is now a G string, giving us B natural instead of flat. The courtesy C natural at middle C is because the G string is unaltered, and Cs played on it should be natural.
But it is true that it was common in the baroque period to indicate accidentals "redundantly" across multiple octaves. Here's Bach's handwritten E major key signature, with F# and G# indicated in the lower octave as well:
While the other answers (referring to scordatura) are obviously valid, I'd like to add a small insight about early key signatures.
Using single sharp/flat symbols that are implicitly valid for all octaves is a convention that became a standard only after the 17th century, also coinciding with the gradual adoption of the tonality system we currently use.
Until the first half of the 18th century, few keys (and related signatures) were actually used, rarely going beyond 2 or 3 symbols. Therefore it was common to indicate "all" sharps or flats for the instrument/voice range.
See the following example taken from the related Wikipedia article:
Also, early key signatures were gradually introduced, and initially with just one flat (no sharp based keys), and even when more "keys" were introduced, there were still a few actual keys commonly in use.
The common flat minor keys (D, G and C) were also often notated with one flat less, being it the sixth grade, therefore actually making them using the Dorian mode, which was traditionally considered the "first mode".
Take this excerpt from the chorale "Die güldne Sonne" (BWV 451), which is clearly in C minor, but is missing the A flat in the signature:
Similarly, major keys (G, D, A) were sometimes omitting the last sharp, the leading tone.
Initial flat keys also used the signature for the lower voice only ("partial signature"), therefore mostly using the Lydian mode except for the lower part: there were combinations of B♮ and B♭ (from top to bottom), caused by contrapuntal polytonality in which the melody tends towards the B-natural (Lydian) tonal realm, while the lower parts tends toward the B-flat one (as reported from the Harvard Dictionary of Music).
This may seem the case of the example at hand, due to the lower C (the leading tone, and last sharp in the D key), but it's just an interesting coincidence caused by the specific scordatura combination used there and the relation between the tuning of the strings (one fourth and two fifths).
I believe: This is not standard music notation. I don't know how scordatura should be written, but it should be interpreted as follows.
A good option is to notate it with a 2# key signature and natural marks on every C#4:
It is even more 'normal' to the modern musician if everything is written in 2#:
