Extract Local Maxima

Python, 54 bytes

f=lambda l,*p:l and l[:p<=l[:1]>=l[1:2]]+f(l[1:],l[0])

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I/O is with tuples rather than lists.

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Python, 57 bytes

f=lambda l,p=0:l and l[:[p]<=l[:1]>=l[1:2]]+f(l[1:],l[0])

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Alt 57:

f=lambda l,p=0:l and l[l<[max(p,*l[:2])]:1]+f(l[1:],l[0])

Mathematica 22 Bytes

Pick[#,MaxDetect@#,1]&

Haskell, 50 49 42 bytes

f l=[j|i:j:k:_<-scanr(:)[0]$0:l,k<=j,i<=j]

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scanr(:)[0] makes a list of the tails of (0:l), each with a final 0, e.g. for l = [4,3,3,4]: [[0,4,3,3,4,0],[4,3,3,4,0],[3,3,4,0],[3,4,0],[4,0],[0]] which is pattern matched agains i:j:k:_ to extract all lists with at least 3 elements which are named i, j, and k. Keep j if its >= i and j.

Edit: Ørjan Johansen saved 7 bytes. Thanks!