The IHIH Pyramid

Python, 165 145 133 123 bytes

A recursive solution:

def i(e):
 d="|";a=e*2;x=d+" "*(a-1)+d
 if e<1:return d
 if e%2:d,x=[" ","-"*(a+1)]
 return[x]+[d+z+d for z in i(e-1)]+[x]

Called with print ("\n".join(i(int(sys.argv[1])))), where the parameter is the iteration number of the IHIH pyramid.

Thanks to @DJMcMayhem for saving 20 bytes. Taking the idea behind those suggestions further saved another 12 bytes. Thanks to @Maltysen for suggestions that trimmed some more bytes.

The function sets the delimiter d to "|" and the intervening spaces to " " (for odd-numbered iterations), deals with returning in the degenerate case, then resets the delimiter to " " and the intervening spaces to "-" for even-numbered iterations. The function returns a list of strings for each line of the IHIH, having embedded the result of a recursive call to the function in the right place within the list.

Cheddar, 186 177 165 154 148 131 bytes

(n,b?,c?,q?,g=s->(n-=1)<0?s:g((q=(c=s.lines[0].len)%4>2?b='|'+" "*c+"|":b='-'*(c+2))+"\n"+s.sub(/^|$/gm,q?'|':' ')+"\n"+b))->g("|")

Uses recursion. Will add explanation once done golfing.

Try it online!

Explanation

This one is a bit complex too keep track off all the variables I'm using but I'll try to keep it simple:

(
 n,    // Input
 b?,   // Stores row to add to top/bottom
 c?,   // Width of string 
 q?,   // false if I-ifying. true if not
 g=
   s->          // Main logic, s is generated string
    (n-=1)<0 ? s :   // Decrease input each iteration. Stop when 0
    g(               // Recurse with....
      (
        q= (         // Set `q` true if h-ifying. false if I-ifying
         c=s.lines[0].len    // Set `c` to width of string
        ) % 4>2 ?
        b='|'+" "*c+"|" :    // Set `b` to top/bottom row adding
        b='-'*(c+2)          // `*` is repeat, c is from before
      ) + "\n" + 
        s.sub(/^|$/gm,       // Add the following to beginning/end of each line
          q?'|':' '          // if H-ifying, add `|`s if I-ifying add spaces
        ) + "\n" + b         // Add bottom row, generated from before
    )
) -> g("|")     // Middle item is `|`

This was a pain to golf but its 55 bytes shorter than original.

Python 2, 93 bytes

Leaky Nun saved 7 bytes.

r=range(input()+1)
r=r[:0:-1]+r
for y in r:print''.join('| -'[[x%2,y%2+1][x&-2<y]]for x in r)