As others have mentioned this is answered a lot on the forum. But I can't help myself when it comes to trying to make bash do this sort of thing (sequences will have to be linearised).

#!/bin/bash

i=1;
while read line ; do
  if [ ${line:0:1} == ">" ] ; then
    echo "$line" >> seq"${i}".fasta
  else
    echo "$line" >> seq"${i}".fasta
    ((i++))
  fi
done < $1

Usage:

$ bash splitfasta.sh multifasta.fasta

Disclaimer:

You should always use a proper parser though (like biopython) as it'll catch many of the special cases. My code just has the bonus of not requiring anything to be installed to run.

This is a python script for splitting FASTA file into an individual file.

from Bio import SeqIO
import argparse

parser = argparse.ArgumentParser(description="Split the fasta file into individual file with each gene seq")
parser.add_argument('-f', action='store', dest='fasta_file', help='Input fasta file')
result = parser.parse_args()

f_open = open(result.fasta_file, "rU")

for rec in SeqIO.parse(f_open, "fasta"):
   id = rec.id
   seq = rec.seq
   id_file = open(id, "w")
   id_file.write(">"+str(id)+"\n"+str(seq))
   id_file.close()

f_open.close()

To run above code, (save the above code in code.py file)

python code.py -f fasta_file

Note: You need to install Biopython module SeqIO to run this code.