Help me I'm lost in the ocean!

05AB1E, 13 12 10 bytes

Saved 2 bytes thanks to Emigna.

Since there is no trigonometric functions to be called using OP's assumption that the earth is locally a plane, it becomes possible to make an 05AB1E solution.

•1#oC•+¹*t

           For understanding purposes input is referred to as 'h'
•1#oC•     Push 12742000 [ = 2*R, where R is the radius of earth]
      +    Compute 2*R+h
       ¹*  Multiply by h. Result is (2*R*+h)*h
         t Take the square root of it and implicitly display it

Try it online!

Python, 34 26 bytes:

(-8 bytes thanks to Osable!)

lambda i:(i*(i+12742))**.5

An anonymous lambda function. Takes input in kilometers and outputs in kilometers. Invoke as print(<Function Name>(<Input>)).

PHP, 34 bytes

<?=sqrt((12742e3+$h=$argv[1])*$h);

breakdown

   r^2+x^2=(r+h)^2      # Pythagoras base
   r^2+x^2=r^2+2rh+h^2  # right term distributed
       x^2=    2rh+h^2  # -r^2
       x =sqrt(2rh+h^2) # sqrt

so far, this is identical to the old Mathematica answer

sqrt(2*6371e3*$h+$h*$h) # to PHP
sqrt(14742e3*$h+$h+$h)  # constants combined
sqrt((14742e3+$h)*$h)   # conjugation to save a byte
((14742e3+$h)*$h)**.5   # same size

now all that´s left to do is to add input =$argv[1] and output <?= - done