Row reduction over any field? EDIT: as stated in the first answer, my initial question was confused. Let me restate the question (I have to admit that it is now quite a different one):
Let's say we have a matrix $A$ with entries from $\mathbb{C}$. Such matrices form a vector space over $\mathbb{R}$ and also form another vector space over $\mathbb{C}$. If we want to row reduce $A$, we see that often we cannot row reduce it over $\mathbb{R}$, that is, using elementary operations involving only scalars which are real numbers, however we can row reduce it using operations that involve complex numbers. 
In conclusion, it seems to me that the row reduction of a matrix with elements from a field (all those matrices form a vector space) may or may not be possible, depending on the underlying field of that vector space.. So this helpful tool (row reduction) is not always available for us to use when we get a little more far from, let's say, the most "elementary" vector spaces.
Is my observation correct? 
Question as initially stated (please ignore):
Consider the vector space of complex matrices over the field of (i) complex numbers and (ii) real numbers. It is straightforward to find examples where a complex matrix can be row-reduced (say, to the identity matrix) in case (i) but cannot be row-reduced in case (ii). 
What gives? So, we can assume that a matrix may be invertible over $\mathbb{C}$ but not over $\mathbb{R}$? 
Thanks in advance..

Your question is confused. What would it mean for a non-real complex matrix to be invertible over $\mathbb R$? That it has an inverse in the set of real matrices? This seems unlikely, as the inverse of a real matrix is always real, if it exists, and being inverse matrices is a symmetric relation.
