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alex@alex-laptop:~$ python3
Python 3.4.3 (default, Mar 26 2015, 22:03:40) 
[GCC 4.9.2] on linux
Type "help", "copyright", "credits" or "license" for more information.
>>> infj = complex(0, float("inf"))
>>> infj
infj
>>> infj*1
(nan+infj)
>>> infj*1*1
(nan+nanj)

I've found the same behaviour going back to Python 1.5.
I think what happens here is that (0+∞j)*1 evaluates the multiplication by implicitly coercing the int to a complex:
 (0+∞j)*(1+0j)
 => (0*1 + ∞j*1 + 0*0j + ∞j*0j)
 => (0-NAN + ∞j+0j)
 => (NAN + ∞j)
rather than using the "simple" way (1*0 + 1*∞j) => (0+∞j).
The problem here is that while there is no mathematical difference between multiplying by 1 or (1+0j), once you involve NANs and INFs, it does. So even though they give different answers, both methods are correct, for some value of "correct".
I don't see that this is necessarily a bug to be fixed, it might count as a change in behaviour needing to go through a deprecation period first. Perhaps it should be discussed on python-ideas first?
My personal feeling is that Python should multiply a complex by a non-complex in the "simple" way, rather than implicitly converting the int to a complex. Anyone who wants the old behaviour can easily get it by explicitly converting 1 to a complex first.
So I guess this is a +1 to "fixing" this.
(Oh, the same applies to the / operator.)

There's no bug here: as Steven explained, this is simply the result of the usual arithmetic conversions when performing a mixed-type operation.
> Python should multiply a complex by a non-complex in the "simple" way
I think this would just be adding unnecessary complication. -1 from me.

Closing as not a bug; for the suggestion of changing the mixed-type arithmetic behaviour, I think that's something that should be discussed on the python-ideas mailing list.